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[|lrttnrr I ftttnn[tR HARPTFI N E W Y O R K L O N D O N T O R O N T O


Tomy parents, andLeonard, theirendless SaIIy loae, for guidance, encouragement. snd

A universal beauty showed its face; T'he invisible deep-fraught significances, F1eresheltered behind form's insensible screen, t]ncovered to him their deathless harmony And the key to the wonder-book of common things. In their uniting law stood up revealed T'he multiple measures of the uplifting force, T'he lines of tl'reWorld-Geometer's technique, J'he enchantments that uphold the cosmic rveb And the magic underlving simple shapes. -Sri AurobitrdoGhose (7872-1950, Irrdinnspiritunl {uide, pttet ) Number is the within of all things. -Attributctl to Ptlfltngorus 580-500s.c., (c. r nrr t Greek Tililosopthe d rtutIrcnutt i cinn) f'he earth is rude, silent, incomprehensibie at first, nature is incomprehensible at first, Be not discouraged, keep on, there are dir.ine things well errvelop'd, I swear to vou there are dirrine beings more beautiful than words can tell. -W nl t W hi tmm fl 819-1892,A meri cnn noet) Flducation is the irrstruction of the intellect in the laws of Nature, under which name I include not merely things and their forces, but men and their ways; and the fashioning of the affections and of the r,t'ill into an earnest and loving desire to move in harmony with those laws. -Thomas Henry Huxley ( 1B25-1 Bg Ett glislthiologist ) 5,





Acknrwled I'd like to expressmy sincere appreciation and thanks to the good people who produced this book at HarperCollins, especiallymy hardworking editots Rick Ko! Alice Rosengard, and Eamon Dolan, and copy editor JeffSmith. I'd also like to thank Nancy Singer for helping celebrategeomefry in the way she designed the bool and Gail Silver for her insightful page layout. And many thanks to my astute agents, Katinka Matson and John Brockman. I especially wish to express my gratitude to John Michell, whose books, lectures, friendship, intelligence, humor, and vision have inspired me and manv others to explore the harmony in nunibers and shapes and *ho generously gave this book its main title and preface. The works of many peoplefrom the deeppast and immediate present have impassioned my interest in this subject, and I would like to acknowledge my debt to many including Keith Critchlow, John Anthony West, Robert Lawlor, Matili Ghyka, D. W. Thompsory T. A. Cook, ]ay Hambidge, M. C. Escher, R. Buckminster Fuller, David Fideler, Iean and Katherine LeMe6, Rupert Sheldrake, Rachel Fletcher, R. A. Schwaller de Lubicz, Sri Aurobindo, Pythagoras, and plato, as well as the artisans and architects of ancient Egypt and Greecewho knew how to make harmony visible. For years of deeply interesting conversatiory fun, friendship, love, and support I'd tike to offer warm thanks and love to my brother, Jeffrey,and my parents and to my friends who live in the canyons on that island off the mainiand, New York City, especially Naomi H. Cohen, fim pittman, Libby Reid, BarbaraCrane,Charlie and Evelyn Herzer,Mark Haj-



at seldis, and June Cobb of the SacredGeometry Library Se C"utn"at"f of St.lohn the Divine. To my friends embracedby the wide sky and natural order of Baker,Nevada' hearttelt ttlut*t fo. tfn" perpetual welcome and the tools with which to leam, i"ctudi"i naryn M. DeBit, Anita Hayden,-and Jim Dalton. I'd also hle tothank my. friends, fellow educators' and students for sharing a dozen lovely years in the green- ' erv of Gainesville,Florida. I'd liku to ."^umber my teachers,who tried as best they could to get me to like math, and my many students,in geometryivorkshops, from whom I have leamed so much' " Andi wish to eipress my wonder and thanks to nature' whose geometric jewelry adoms the world'

and fienmetry flealit thefluestfur lohn Michell Sooner or later there comes a time in Me when you start thinking about Reality and where to find it. Somepeople tel l you there is no such thing, that the world has nothing permanent in it, and, as far as vou are concemed, consists merely of your fleeting experiences. framework, they say, Its is the random product of a nafural process,meaningless and undirected. Others believe that the world was made by a divine Creator,who continuesto guide its development.This soundsa more interesting idea, but, as skeptics point out, every religion and church that upholds it does so by faith alone. If you are naturally faithful and can accept without question the orthodoxy of your particular religion or system of beliefs, you will feel no need to inqufue further and this book will appear superfluous. It was written for those of us who lack or have lost the gift of simple faith, who need evidence for our beliefs. We cannot help being athacted by the religious view, that the world is a harmonious, divinely ordered creation in whidr, as Plato prornised the uninitiated, "things are taken care of far better than you could possibly believe." Yet superficially it is a place of confusion and chaos, where suffering is constant and the ungodly flourish.



This is where we begin the quest for Reality' Lgokinq closelv at nature, the firsi insight we obtain is thai, behind the apparently endlessproli{eration of natural objects,there is a flr lesser number of apparently fixed tlpes' We see' for example,that through every generationcatsarecatsand are progrlmmed for catlike behavior. In the same way, every 'ro"""hu" the unique characteristics of a rosb and every oak leaf is definitely an oak leaf. No two specimens of these -are ever exactlythe same,but eachone is clearly a product of its formative i1pe. If it were not so, if animals and plants,simthe ply inherited their progenitors' characteristics, order of soon dissolve into an infinite variety of creanature would tures, undifferentiated by speciesand kinships' This observation, of one type with innumerable products, gives rise to the old philosophical problem o{ the One and the Manv. The problem is that, whereas the Many are visible and tangible ind can be examined at leisure, the One is never seen or sensed, and its very existence is only inferred tfuough the evident effect it has upon its products, the Many. Yet, paradoxically, the One is more truly real than the Many. In the visibte world of nature all is flux' Everything is either being bom or dying or moving between the Nothing ever achievesthe goal of perfection two processes. ti'," ttute of equilibrium that would allow it to be o. The phenomenaof nature, said Plato, describedin essencb. are always "becorning," never actually " are." Our fiue sensestell us that they are real, but the iniellect judges differently, reasoning that the One, which is constant, creative, and ever the sarne,is more entitled to be called real than its ever-fluctuating products. The search for Reality leads us inevitably toward the type, the enigmatic One that lies behind the obvious world of*re Many. Imrnediately we encounter difficulties. Being imperceptible and existing only as abstractions, tyPes canA not be apprehendedby the methods of physical science. number of modem scientists, perceiving the influence of types in nature, have attempted to bring them within the range of empirical study. Rupert Sheldrake, author of A Nezu Science Life and other works, has taken a bold step in that of worked on sysdirection.In an earlier age,the Pythagoreans of numerical formulae. Yet tematizing the types by means


Plato, who wrote at length on the subject of constant types (referred to as "forms"), was carefully ambiguous in defining them and never made clear the means by which they influence the world of appearances. Plato did, however, give instructions on the procedue toward understanding the nature and function of the types. In the Republiche described the ascent of the mind through four different stages.It begins in Ignorance, when it does not even know that there is anything worth knowing. The next stage is Opinion, the stage in which TV chat-show particiDants are forever stuck. This is divided into two subcategories, Right Opinion and Wrong Opinion. Above that is the Ievel of Reason.By education and study, particularly in certain mind-sharpening subjects,the candidate is prepared for entry into the fourth stage, which is called Intelligence (nous). One can be prepared for it but with no guaranteeof success,for it is a level that one can only achieve on one's own, the level of heightened or true understanding, which is the mental level of an initiate. The studies that Plato specified as most effective in preparing the rnind for understanding are the so-calied mathematical subjects, consisting of number itself, music, geometry, and astronomy. These were the main studies of Plthagoras and his followers, who anticipated the realization of modem physics in proclaiming that all scales and departrnents of nature were linked by the same code of number. Geometry is the purest visible expression of number. In Platonic terms, the effect of its study is to lead the mind upward from Opinion onto the level of Reason,where its premises are rooted. It then provides the bridge or ladder by which the mind can achieve its highest level in the realm of pure Intelligence. Geometry is aiso the bridge between the One and the Many. When you draw one of its basic figures-a circle, say, or a triangle or regular polygon-you do not copy someone else's drawing; your model is the abstract ideal of a circle or triangle. It is the perfect form, the unchanging, unmanifest One. Below it are the Many-the expressions of that figure in design, art, and architecture. ln nature also the One circle gives rise to the Many, in the shapes and orbits of the planets, in the roundness of berries, nests, eyeballs, and the



cycles, of time. On every scale, every nafural pattern of growth or movement conforms inevitably to one or more of the simple geomerrictypes.The pentagoryfor example,Iies behind each specimen of the five-pelaled rose, the five_ fingered starfish, and many other living forms, whereas the sixfold, hexagonal gpe, as seen in the structure of snowflakes and crystal generally, pertains typically to inan_ imate nature. As soonasyou enter upon the world of sacred, symbolic, or phiJosophicai geometry-from your first, thoughifuJcon_ struchon of a circle with the circumferencedivid;d into its natural six parts-your mind is opened to new influences that stirnulate and refine it. you begin to see,asnever before, the wonderfuily patlerned beauty-of Creation. you see true artistry, {ar above any human contrivance. T?risindeed is the very, sourceof art. By contact with it your aestheticsensesarc heightenedand set_upon firm bisis of truth. Beyond the the oovlous pteasureot contemplatingthe works of nafure_the MTy-1" the delight that comes through the philosophical study of geometry, of moving toward-the prr".,." of th" One. Michael Schneideris an experienced teacherand, as you areentitled to expect,a masterof his geometriccraft. No Lne less qualified could set out its basic piinciples so ctearty anJ simply. His much rarer assetis appieciation of the symbolic and,cosmologicals].'rnbolisminjrerent in geometryiThat is the best reasonf9r being interestedin the subject,and it is why. rhe.philosophers of ancient Gieece,Egypt, 1"_,t:i:." and other civilizations made geometry and numbei'the most important of their studies. The traditional science taught in their mystery schoolsis hardly known todav.Itis not available for study in any modem place of educition, and the.reis very little writing on the sublect.ln this book you will find something that cannot be obiained elsewhere, a complete introduction to the geometric code of nature, written and illustrated by the most perceptive of its modem investigators.


Thr*ort thing incompr ehensible abouttheuniaerse that is it is comprehensible. -Albert Einstein (1,879-1955)

Nothing in education is soastonishing the as amount ignorance of it accumulatesthe in form of inert facts. -Henry Brooks Adams (183&-1918, American historian)

The universe may be a mystery, but it's not a secret.Each of us is capable of comprehending much more than we might realize. A vision of rnathematics different from that which we were taught at schoolholds an accessible to a nearby key world of wonder and beauty. In ancient Greecethe advanced students of the philosopher Pythagoras who were engaged in deep studies of natural science and self-understanding were called,mathematekoi,a'thosewho studied all.l' The word mathemasigntfied "leaming in general" and was the root of the Old English mathein,"to be aware," and the Old German munthen, "to awaken." Today,the word math has, for most people, constricted its scope to emphasize mundane measurement and mere manipulation of quantities. We've r;nwittingly traded wide-ranging vision for narrow expertise. It's a shame that children are exposed to numbers merely as quantities instead of qualities and characterswith distinct personalities relating to each other in various pattems. If only they coirld seenumbers and shapesas the ancierrtsdid, as s)zmbols principles availableto teachus about the natof urai structure and processesof the universe and to give us perspective on human nature. Instead, "math education" for children demands rote memorization of procedures to get one "right answer" and pass innumerable "skill tests" to prove superficial mastery before moving on to the next isolated topic. Teacherscall this the "drill and kill" method. Even its terrninology informs us that this approach to math is full of problems. It's no wonder countless people are irmumerate. We've lost sight of the spiritual qualities of number and shapeby emphasizing brute quantity.


This book concemsmathematics,but not the kind you were shown at school. The Roman goddessNumeria is said to have assistedin teaching each child to count' We must have had a misunderstanling becauseI grew uP on uneasy terms with the subject of irathematicJin schbol. I was intrigued b.y.the inJinity of numbers and could calculate mechanically if I had when Bodily ,rrrcise, I was taught for a given situation' does compulsory, no harm memorized the rule Although I liked science and art, math intimidated me' I to the body;but knowledge remem6er my frustrated tears at age seven over the concePt under tohichis acquired of subtracting by borrowing {rom another column when it no contained a zero.l dreaded math, its mindless memorizaobtains compulsion tion and its tedious paperwork' (Pify the teacherswho have holdon themind. to check it!) Math was dry and mechanical and had little rel-Plato (c. 428-348e.c.,Greek evanceto my wo d. If we had iooked at numbersto seehow they behave with each other in wonderful patiems I might mathematical philosopher) have liked math. Had I been shown how numbers and shapes relate to the world of nature I would have been ihdlled. hstead, I was dulled by math anxiety and pop ottizzes. Fortunately for me, when I was sixteen one of my teachers mentioned that mathematics can be found in nature: a a bee's cell and quartz shaped Wrao* shouldbe six-sided snowflake is to make like cormection. How could the I was stunned crystal. as of cherished a means something as apparently irrelevant as mathematics be youth to old related to something as wonderful as nature? The ordinary traaeling from world opened up to rne and spoke the language of number age, it is morelasting for mathematics thanany otherpossession. and shape. No longer a foe, the dreaded a teacherand a tool. Nature wasn't what it becameat once -Bias of Priene (c.570B-c-, was made out to be, an antagonist to fear, conquer, and exploit, but a garden of wonder and a patient teacher worof one of the SevenSages ancient Greece) thy of gteat respect. Over the years rny studies led me to seethat a profound understanding of number was prevalent in ancient times, more than is commonly acknowledged, and seamlessly wove mathematics, philosophy, ar! religiory myth, nature, science, technoloW, and everyday life. This book is a fanning of some little sparks of philosophy from deep antiquity to introduce ihe general reader to another view of mathematics, nafure, and ourselves that is our heritage and birthright. It requires that we liberate math from its Pigeon-


Cqlvinona Hobbes '(EAl{. A\! 1tl6E EAUAnoN: IHls ul\{nE Bcd( ts N\! OFII{INGSTHAT AF L\KEMIRAC,LES, \C'\) lb HA\IE IAG TTIO N\}USgSA\OiI\ITN E. A<PIF' OI\ FAII\{I YCN l{EM, TI MAGI(A$ IT5 A. AOD &cch4EoNEr0ty NUMER REL\GION ! I NOONECANSA\ I{ONIf HARENS. EIII.$R \C|\} EEL\BIE r oR.\cN Dor\T.

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hole and see it spanning the framework of the most interdisciplinary topic possible:the uliverse. Plato wrote that all knowledge is already deep within us, so no one can really teach us anything new. But we can remind each other of the archetypal principles of number and nature we already know but may have forgotten. This book is intended as that reminder to guide you through this world of wisdom, beauty, uplift, and delight and to remind you of the gentle, wise principles by which the universe is designed.

By *rr* [theGreeks] geometrywasheldin the aeryhighest honor, and noneweremoreillustrious than mathemati cians.But we [Romans] haoe limited thepractice this art to of its usefulness in. measurement and calculation. -{icero (106-43 0.c., Roman orator, statesman, and philosopher)

THREE LEVELS OF MATHEMATICS Mathematics is whole, but it rnay be divided into three levels or approaches: secular,symbolic, and sacred. Secutor Mothemotics What is taught in schoolcan be called secularmathematics. Adding the amounts of a storc purchase,, calculating change, weighing produce, measuring ingredients for a recipe, counting votes, telling time, designing a bookshelf or skyscraper,measuring iand boundaries, stock market economics, calculating tax: we're taught ihese are the ends of mathematics for nonscientists. We train children to be hurnan pocket calculators.Even many educated people who are

Ilature, thatuniaersal andpublicmanuscript. -Sir Thomas Browne (1605-1682,English physician and author)

adept at calculating have no idea that there is more to mathematics than mere reckoning of quantities. This quantitative approach keeps us duII to the potential wisdom that the familiar counting numbers can teach us. \ltrhen imaginativeiy taught to people begiming at an early age, mathematics can delight, inspire, and refine us. It can make us aware of the pattems with which the world and we are made. Instead, math is taught as a servant of co [nerce, without regard for its basis in nature. It is viewed as a distant subiect that instills much more anxiety than wonder and inspiration. Mathematics is seen as outside us to be occasionally caiied upory rather than woven into the fiber of our existence. SUmbotic Mothemotics

Nature is writtenin and symbols signs. -John GreerLleaf\A4rittier (1807-1892, American poet)

AII thingsare full of signs,and it is a wise man whocanlearnaboutone thingfrom another. -Plotinus (205-270,Roman Neoplatonic philosopher)

Embedded in the very structure of ordinary numbers is another level, which may be called "symbolic" or "philosophical mathematics." It is well known that simple numbers and shapesrelate to eachother in harmonious recurring patterns. Just notice how floor tiles containing different shapesmesh. Mathematicians and scientists seek and study patterns as clues to a deeper understanding of the universe. What's more, s).'rnbolic mathematics recognizes mrmbers and shape patterns as representative of far-reaching principles. They can be guides to a deep cosmic canon of design. Nature itself rests on an intemal foundation of archetypal principles symbolized by numbers, shapes,and their arithmetic and geometric reiationships. According to ancient mathematical philosophers, the simple counting numbers from one to ten and the shapes that represent them, such as circle, line, triangle., and square express a consistent, comprehensible language. The ten numbers are a complete archetypal sourcebook.They are the original ten patents for designs found all tfuough the r.rniverse. These ideal oattems are the ones that were skewed and veiled in schobl and that nature approximates in all transitory forms, from the smallest subatomic particles to largest galactic clusters, crystals, plants, fruits and vegetables, weather pattems, and animal and human bodies. Any-



Islamic tiling pattem of a mosque and structure of the boric acid molecule show how identical shapesinterlock in defined, recurring pattems in both art and nature.

I am not ambitious to appeffi manof letters: a I could becontenttheworld shouldthink I hadscarce Iooked upon any otherbook than that of nature. thing anyone can point to in nature is composed of small pattems and is a part of larger ones. Historically, nature has been compared to a book written in geometriccharacters. 1611Calil-eowroie: In Philosophyis written in this grandbook-I meanthe universe-which standscontinually open to our gaze,but it cannotbe understoodunlessone fust leams io comprehend the language interpret the characters whiih it and in is written. It is written in the languageof mathematics, and,itscharacters triangles,circles,and other geomet_ are rical figu.res, without which it is humanly impos'sible to understand singleword of iq without these, is wan_ a one dering aboutin a dark labyrinth. Reading the Bookof Naturc hrst rcquires familiaritv with its alphabet of geometric glyphs. WeG exposed to nature,s text in the natural shapesaround us, but we don,t recosnize it as something we can "read.,,Identifying shapesanjoat_ tems and knowing what principles they relresent allow'sus to understand what nature is doing in any given situation and why these principles are appUea in numin affairs. Why areplatesand pipes and planeiJ round? Why do hurricane's and harr-curls/atoms and pineconesunfurl as spirals?Whv -Robert Boyle (1627-1,691, British physicist and chemist)


T In nature's infinitebook of secrecy,littti I canread. a -William Shakespeare (1,564_L616)

Lioing in theworld zaithout insightinto the . hiddenlaws of nature is like noi knowing the language the country in of whichonewasbbrn. -Hazrat Inayat Khan (1882-1922 Sufi spiritual guide, musician, and writer


didn't school teach us that when we see the same spiral shape in shells, galaxies, and watery whiripools we are witnessing the principle of balance through motion? Ol that the hexagonal cells of beehives package the maximum space using the least materials, energy, and time? Nature labels everything with a cosmic calligraphy, but we generally don't suspect even the existence of the language. It is an open of For thenature number secret,fully in view but usually ururoticed. Like consonants and vowels-like building blocks and Srowth Patternsis to beinformatioe, numbers, shapes,and their patterns s;'rnbolize omnipresent guiding and instructiae for principles, including wholeness, poladry shucture, balance, that in anybody eaerything cycles, rhythm, and harmony. Each shape represents a difto is subject doubtand thnt ferent problem-solving strategy in the cosmic economy. To is unknown. see more deeply into this design alphabet we must be conversant in nature's native tongue, the language of symbolic For nothingabout mathematics. This book is primarily concerned with symthingswouldbe bolic mathematics.

to comprehensible anybody,neither of things nor in themselaes, of onein relatian to the other,if number and its essence werenonexistent... Theessence number, of Iike harmony,doesnot s allow misundertanding, to for this is strange it. Deception enr'yare and to inherent theunbounded, unknowable, and unreasonable.. kuth, . howeuer, inherent the is in natureof numberand inbredin it. -Philoiaus (Fifthcentury e.c.,Greek Pythagorean philosopher)

Socred Mothemotics The terms "sacredsites,""sacred geography,""sacredarchitecture," "sacred arithmetic," and "sacred geometry" seem overused today. To the ancients the "sacred" had a particular sigrrificance involving consciousnessand the profound mystery of awareness. How are you and I aware of these very words and their significance? Now that you've read them, these words are no longer just on the page. They're within your awareness.Sacredspaceis within us. Not in out body or brain cellsbut in the volume of our consciousness\Arhereverwe go we bring the sacred within us to the sacred locationsand studiesby the Presaround us. We consecrate ence of this awareness,not just the other way around. \44ry should the sites of stone temples and wonderful cathedrals be more sacredthan a rocky desert or concretecity streetif we bring holy consciousnessto each of them? Geographic locations are no more or less sacredthan any other, although they may be powerful telluric sites. Awareness is the ultimate sacred wonder. \4rhy endow objects outside ourselves as sacred and ignore the same source within us? A surprising amount of the world's religious art and architecture has been designed using the timeless symbolic pattems of


nafure and number, but thesepattems remain syz bolicof our own sacredinner realm, s).'rnbolic of the subtle structure of awarenesswhose source is the same as archetypal numberAll this was understood in ancient times ani'deemed so important that ii was built into the culture on every level. Modem science tells us that what we comrnonly call "rcality" is a compilation of pictures based on u ,rirro* sense-bandview of surface features. The world we perceive is a small slice of a vast, mostly invisible energj,-eve.,t. Mathematics can take us beyond our ordinary linriis to the cosmic depths. Plato at his Academy requirej the studv of mathematicsas a prcrequisitefor pftito{oinA,u term sigriifying "the love of wisdom" and "to lift the soul to truthl Iu;t over a centuq/ earlier Pythagoras had invented the word "p_hngsophy" as a result of a question posed to him. I4rhen asked "Are you wise?" he is said to have answered ,,No, but l'n:. a loaer of wisdom." Both pythagoras and plato sug_ gested that all citizens leam the properties of the first ten numbers as a form of moral instruction. The study and con_ templation of number and geornetry cnn show us, if we look wi$ the eyes of ancient mathematical philosophers, that neither outer nature nor human nature is the hodgepodge it may seem. Symbolic mathematics provides a map of our own irurer psychological and sacred spiritual But studying number properties and intellectually knowing the road map, the symbolism, is not the sameas achrallv ta*kine the joumey. We take that joumey by finding within ourl selves the universal_ principles these properties represent and by applying the knowledge to our own growth. We pay lltention to paying attentiory in imageless u*u.".i"rr, directing sustained attention to that whici the srrmbolsrefq to within us. lA/henthe lessonsof symbolic or philosophical mathematics seenin nature, which were designed into reli_ gious architecture or art, are appliedfunctioiatty (not iust inteffectually)fofacilitatethegrowthani transformitiin of ionsciousness, mathematics then may rightly be called ,,saired.,, To me, the terms "sacred arithmetic;and ,,sacredgeometry,, onJyhave significancewhen grounded in the exieriencetf selJ-awareness.Religious art is sacrednot only due to its subject matter but also becauseit was desigrredusine the subtle s).'rnbolic languageof number, shape,Ld propoition

amuse me,"I said, "with your oboious fear thnt thepublicwill disapprooe thesubjects if you prescribe don't seem usefal.But it is in fact no easymatteLbut ztery dfficultfor people to belieae that thereis a faculty in the mind of each of us which these studies purify and rekindleaffer it hnsbeen ruinedand blindedby otherpursuits, thoughit is moreworth pleservingthan any eye sinceit is theonly organ by which we perceioe the truth. Those who agree with us aboutthis wiII gioe y our proposals unqualifed i approaal, those but who arequite unawnreof it will probablythink you are talking nonsense, they as Toon't whatother see benefitis to beexpected from suchstudies."




to teach seLf-understanding and functional self-developrnent. Ancient Egyptian arts, crafts, and architecfure Perthat to not a problem besolaed, haps provide tnJ"Uest accessible examples of design .rc"d itt symbolism of nurnber, geometuy, and nature to it is a realitYto be to form of self-development trained initeachan accelerated experienced. tiates who knew how to translate the symbolism into meditative exercises. -J. J. Van der Leeuw Thus, true sacred Seometry cannot be taught through books,but must .u*uii as puti of the ancientoral tradition passed from teacher to pupil, mouth to ear. Becauseover ienturies this knowledge was passedalong secretlyso asnot to conflict with the prevailing intolerant religious authorities or be disposed to those considered profane, there is still Th, gootof tifeis liaing in an aura of mysticism about it. But nature's Patterns and with agreement nature. those o{ our inner li.fe are familiar to everyone and always available to us. The power which we seek is the p ower with -Zeno of EIea (490-430 8.c., which we seek. When we feel separate from the archet'?es Greek philosoPher) of nature, number, and shape we make them mystical, but this orilv keeps our selves in a mist. There is no need for ';occultism" any longer. These are everyone's secrecyand iife-faits. We can apply them to better appreciatethe world, and we'Il need thern once we realize the urgency of cooperating with the way the world works. This book is concemec with dispelling the mysticism surrounding sacred matheW, orrbornfor matics by reiniegrating the timelessknowledge of number cooperation. and shape wi*i its familiar expreisions in nature, art, and its -Marcus Aurelius(121-180, practical significancefor self-development. philosopherat twelve, Studying, contemplating, and iiving in agreement with R6manemperorat forty) universal principles is a social responsibility and can be a spiritual path. It is becoming clear that when we cooperate when we resist,we struggle. *ith nature'sways we succeed; Imolications for our environmental crises are obvious. Rather than an antagonist, nature can be our teacher to leam from and cooperate with to mutual benefit. To understand nature better,we first need to recognizethe roles of its basic pattems.

of Th, ,rd mystery lifeis

MATHAS CREATIONMYTH MYTHMATICS: To say that ancient seers viewed the universe and themselves in a way different from ours is understatement. Today, we are emerging from the grip of literalism, mere



quantitative measurement, and analysis. The ancients had a more poetic and synthesizing vision. We think we invented mattu but the ancients knew that they were involved in a processof discovery. They personified in stories the forces of nature and gavethem namesof gods,goddesses, nature and spirits. Mythological exploits heiped explain the orisins and ways of familiar events from world creation t"o night an{ day, storms, agriculture, wars, love, hate, joy, sleep, iife and. death, and the full spectrum of naturai una ni -ur, aftars. ttut today we condescend and say that,show, in their childlike way, those pagans attempted to explain the mvs_ teries of the world while lacking our ,,enlightened,, scientific understanding. We think we rel y underitand the forces of nature because have labels,theodes,terminology,exper_ we iments, measurements,advanced formulas, and aJtoundino Divinity cfu cumscribing the technological tools. With our habit of sensitle h,".ulir- *E limits of the universe.ilote have a difficult tirne believing that the ancient mfihopoeic his foot beyond the frame. imagination could have been a valid tool for seriou" kno*f (BibleMorulisee,c. 1250) edge. Yet it is we who have the cruder understanaing. Uy pigeonholing fraglnented facts we miss the whole haimo_ ruous picfure. We grope in a world we consider dangerous, accidental, and chaotic but one that is actually harm"onious and awaiting our cooperation. If only we could seewith ihe eyesof the ancients! Scientists confiim with formulas what ancient seers knlw thrgugh revelation: that the worta,s puttemsand cycles are harmonious when seen as mathemaiical relation_ ships. The ancientsi intuitive revelations were grou.a"aiy the study of nature, numbers, and shapes."ff," u".i"", werS particularly interestedin thi ways g"o;uti. !,1eks recurring harmonious pattems (ihe Greek w3rd harmonia signifies,,fitting togethe/,), aswe seein floor tiles, quilts, and wallpaper patterns, and how forces of nature orchestrate small and large cycles(assatellites in now o.l" worldwide weather "patLrn).'Wh; ;;-;" ::y:ul "things," nouns and discrete objec'ts,the ancient rnathe-mli ical phiJosophers sawprocesses, verbs,transformine outi;;" meshecl harmoniously.ln truth, the ,,buck,,never s"tops. The r-ropt.ranguage retainsthis vision, having no nouns.(i,m not The Ancient of Days design"wearing a shirt,' but ,,I,m shirting.,,) iikewise, ;.;;rl mg the universe through 'cosmos" generally refers to ,,oute-rspace,,,Bui the word geometry.(Ro&efeller Center) derives trom the Greek kosmos (signi$ng,,embrcjdery,,),



which implied not a universe like a huge room fi1led with discormectednoun-things but the orderlinessand harmonyof woven Datterns with which the universe is embroidered and signified the honorable and "right behavior" moves. Kosmos of the whole, the harmonious orchestration of the world's pattems and processes.In this original senseour word "cosmetics" refers not to the nouns involved, the lipstick and rouge, but to the processof bringing the elements of the face into harmony. Our biggest failureis By studying the recurring harmonious pattems inherent patterns. ourfailureto see in mathematics, music, and nature, ancient mathematical -Marilyn Ferguson philosophers recognized that consistent correspondences occur throughout the universe. The Greeks investigated (American writer) ariihmetic by arranging pebbles in various geometric shapes ("figurate arithmetic") and so rmcovered the archetypal pattems and principles inherent in many tyPes of le1ationships. This sfudy developed over centuries from what we call "number magic" to become the modem branch of mathematics called "number theory." Through extensive


a 1 +2

a aa 3

a o o a oaa ao aao aoaa +3

a ao aaa aaaa aaaaa +5





SQUARE NUMBERS Figurate Arithmetic. Simple a angements of pebbles showed ancient mathematicians the pattems inherent in relationships among numbers. For example, any two consecutive "triangular" nurnbers(1, 3, 6, 10,15 . . .) added together always make a "square" number (1, 4, 9, 16,25...).

O 1

oa Oo 4 +3

ooaa aoo aaaa aaa oaaa OOO aaaa 9 +7 l6

aaaoa aaaoo a a aa o aaaao aaaoa 25 +9

a aa 3 +

a aa aoa o -



study of patterns and by intuitive revelation the ancients realized that the structure and pattems of arithmetic and geometry reenact the creating piocesses found ali through nature. In both number and nature they saw the samedivine impress. Strip away all the sensory characteristics of color, texture, tone, taste, and smell from an obiect and onlv number remains as its size,weight, and quantity. Takeaway the leatures associatedwith number and it's all gone. Ordinary numbers and shapes represent eternal verities in a form made comprehensible to us. Mathematics is a philosophical language that reveals the Greek archai, tlrrefowtdatlon principles of universal design and construction. Numerals-, the written s).'rnbols of nontangible numbers, and geometric shapes,are the emblems of these archetypal principles. The archetypes are universal in that they are the iame io everyone everywhereand in every era Nature's harmonious principles are exhibited through . its mathematical relationships. As we leam to read its archl_ typal language, we discover that the topic of the book of nature is a story a my.thicquest about thl processof transformafion. This mathematjcalmyth of the ireative process. this "mythmatics," is not ancie* or new-age but timeless, accessible everyr in age because encodesetemal constants it available to Although the m1th,s outer form adapts to _all. different cultures, it may be rediscovered in any eia bv exarnining simple numbers, basicshapes,and evei-present nature. Thus, people of every historical period can under_ stand the principles of nature,s creating processby imagi_ natively_examining the corresponding arihetypal relatioin_ ships inherent in mathematics, personified in mythology, and depicted in art and culture. This book is intended a;; begirmer's guide to the process of recovering that vision. Lr the view of philosophical mathematiiians, numbers and their associated shapesrepresentstagesin the process ol becoming.Ifhile integrated in the whole, eachhis a life ot rts own and a unique role in the cosmicmyth. The arche_ types of number and shapewere personified in ancientcul_ tures as various gods, goddesses, and world_builders. The classicmyths eJaborated particular aspects the univer_ on of sal principles they representbd. Their intirrelationships and liaisons, which we consider a ,,soap opera,,, transmitted

Theharmony o|theworld is made manifest Form in andNumber, and theheart and soulandall thepoetry

ofNaturalPhilosophy are embodied theconcept in of mathemntical beauty. -Sir D'Arcy Wentworth Thompson (1860-1948, Scottish zoologis! classical scholar)

The world, harmoniousl y confused, Whereorder in oariety we qP2

And where, all things tho' dtfft , AII agree. -Alexander Pope (1588-1744, English poet)


timelesstruths. Though statesanction,as in Egypt, Greece, India, Africa, Chin4 Tibet, Native America, and elsewhere, all aspects of society were organized according to a canon consistent with nature's own structure. The proportions of temples were always designed, situated, and built in accord with the number and shape symbolism representing the An *rr rXrrr, ofNature ternple's deity. Through the process known as " gemaftia," which associatesnumber values with letters of a carefully areonly the mathematical constructed alphabe! as in Greek, Hebrew, Arabic, Syriac, consequences small of a and Egyptian, the names, titles, and atkibutes of deities numberof immutable laws. reveal, to those initiated to the code, their role in the cosmic constructive process. For example, seven was known as the -Pierre Simon de Laplace number of the "virgin" becaule no number below seven (1749-1827,Frcnch enters into (divides) it, nor does it "reproduce" another astronomer and number within the first ten. Through gematria, the Gieek mathematician) letters of the name of the maiden goddess Athena add up to seventy-severy giving us a standard for measuring length, distance. The letters of her epithet, Pallas, add to 343 (which equals 7 x 7 x 7), indicating volume. The letters of her appellanonparthenos("vftgSn") add to 515, and it is no coinToknow all,it is cidence that 51.5 degrees is extremely close to the angle necessary knowaery to within a regular heptagon, the shape with seven equal sides and angles, giving us area. Each "deity" represented a set of Iittle;but toknou.t that archetypal principles that were rnade cornprehensible as oerylittle,one must first mathematicalpattems for practical use as lengih, area,vollcnow prettymuch. ume, weight, duration, and music and were applied to the -the architecture and ritual of deity's temple precinct. Each --eorges I. Gurdjieff deity was responsible for overseeing part of the world,s har(1.872-1949) monious structure in the same way that his or her mathematical pattems did. Ancient mythology was consciously integrated wi*r, and symbolized by, the universal and timeless canon of nafure's mathematics. This book makes practical the teachings of the Pythagoreans and others (including Plato, Theonbf Smpna, Iamblichus, Philolaus, Proclus, and Nichomachus) concemN umbers thehighest ing the numbers one to ten are as a complete frame for explordegree knoutledge. is of lt ing and constructing the fundamental forms of the univirse; These fust ten numbers are like seedsfrom which all subseknowledge itself. qrrert numbers and shapes grow sharing and expanding -Plato their properties. The use of Greek words for these numbei

xxtx princrples, Monad, Dyad, Triad, and so on, as chapter titles, is intended for rnore than numerical order. Theseterms rep_ resent a processof unfolding cosmic qualities and prinii_ ples. Each chapter is devoted io one number and shapi principle. Although we count in sequencefrom one to ten, irese number principles do not only unfold sequentially but inter_ penetrate the universe simultaneously-in a coimic svm_ phony. The cosmic creating process is deep within us and can emerge through our hands with the heip of the three tradrtional tools of the geometer-the compais, the straightedge, and the pencil. To the ancients these tbols represent""dthr"". divine attributeS that designed the pattemi of the world. Just ln ttds century modern scienceis rediscovering what the -their ancients s)rmbolized by these three tools and use. Albert Einstein referred to them as E = mc?,the famous rela_ tionship among the universe,s three ,,tools,, of the rating process: "orfier_ light, energy,and mass(or matter).The ro'ies and motions of the geometer,sthree tools in geometriccon_ squction replicate the universe,s o-., p.o""ss "."-utirrg whereby i<lealpatterns are approximated in natuial'design on aII levels. We will iook at our own constructions on pap-er as.representing universal creations.Eachstep we take syrn_ bolizes a. cosmic motion. By contemplating the essencl of the ten numbers through our geometric construc_ ti9ry, we.yill.gaze directly at the lines"on the face of deep wisdom, the divine substrateof nature artd ourselves.

Man, though seeing, suffered . ftom blindness.. And,forhim,Ifound Numbers, purest the of inoentinns. -Aeschylus (52F-456n.c., Greek tragic dramatist)

G eometry lotozoledge is of theeternally existmt. -Plato

He therefore wishes who to rejoicewithout doubt in regardto the truths underwng phenomena mustknowhow to deoote

ABOUTTHISBOOK Thebesteffectof any book is that ii excites readerto the self-activity. -Thomas Carlyle (1795_1gg1, Scottish essayist historian) and Verifying number principles for ourselves is more impor_ tant than iust reading about them. No matter how conver_ sant we are with the details of any map or restaurant menrL it's more informative to take thi youiney *a ^oru ,ro,rr_ ishing to eat the food. We can discove, to, o*r.t r.r-tt "

himself experiment. to -Roger Bacon (122V1292, Engl.ishphilosopher and scientist)


If you do not restuponthe good , foundationof nature you will laborwith little profit. honorand less whotake their [T]hose for any standard onebut of nature-the mistress all s-IDeary ffiastef in themselaes ztain. -Leonardo da Vinci



timelessnessof nature's creating process by consciously participating in geometricactivities. By-doing geometric constructionson paPet or with a stick in the sind, we are recalling the agelessprocess of creation, replicating with our mind and hands the generative principles by which the universe is evolving. Our construciions wiil dLmonstrate to us the harmonious princiPles at work in the world. S)'rnbolic and sacred mathematics encode subtle experiencei whose purpose is different from that of secularmathematics.They caninvigorate,refine,and elevateus. Our role as geometeriis to discoverthe inherentproPortiorybalance, utd hut-ot y that exist in any situation. The study and experienceof numeric and geometricproportion infuses in uJan appreciationof proportion everywhere.The study of us balanceteaches to recognizeand seeka senseof baiance in our lives. The study of harmony develops our sense ot harmony in all relationships.Actually to seeand work with unity and wholeness in geometry and natural forms, rather than iust read about them, can help abolish our false notion from nature and from each other. It is this of separateness notion that ultimateiy fuels competition for the "goods o{ the earth" and contributes to environmental crises. The material of this book developedover years of study for and through holding public workshops and courses people of widely varying interests,especiallyartists,architects, and educators. \Alhen rny interest in this subject first began, I was sure that someone must have written one book explaining to my satisfaction the relationship between numThe information art, and se1f. bers, shapes,nature, science, seemsso fundamentai, and people have been around for so many millennia to view it, that I was surprised to learn it only existsin books, articles,and fragmentswritten mostly for specializedinterests.I couldn't find the one book I was looking for that tied it a1ltogether, so I wrote it. It seemsthat the best approachto show you what I've leamed is to take you beyond reading and involve you in the already-existing sy'nthesisof geometry, natural science,art, and sel{-understanding through a "hands-on" approach. This book is intended to facilitate these studies in such a way as to see


ourselvesas part of the overall harmony and have the tools to engagein it. It will guide you: r To seehow number and shape srrmbolisrn is already familiar to us through popular sayings, fairy tales, myths, and religious ritual. e To examinethe archetypalprinciples expressed by stmptenumbersand shapes and to let vou verify the natural relationshlps existing arnong them. You'll do this by following step-6y-step" instructions for re-creating the basic geometric constru.tions s)'rnbolizing unfolding universal principles. You will use the tools of the geometer,s creating process-the compass,straightidge, and penc -to make two-dimensional and three_ dimensional models of the principles expressed by the numbers from one to ten. . To seewhere, and understand whv, these numericaI archetypesprecipitate as organic and inorganic forms in nafure and are the ones we relv upon in tectmology and engineering. you will be shown how to apply the geometric constructions to images of familiar natural forms discoverins for yourself the ways nature speaksh", g.o-"ii" language. . To examine examples of art throughout the world that display mathematicaldesign.By applying the same geometnc conskuctions to reproductionsof art in the book (using tracing " paper), you will leam to seehow arch-etvoal mathemaficalprinciples have beenpurposefully and.consistently applied, to a surprisAg aegrei to jewelry, and architecture throughoul thJworld, art _ endowing them with powerful coripositional structure,spnbolic significance, unJ intended psychological effects. Sorneexamples will show the geometry the artist used.Further examplesare left for you to complete and explore based on the geomehic constructions you have learned.


The real aoyage of discoaery consists in not seeking newlandsbut seeingroith new eyes. -Marcel Proust (1.871.-1922, French novelist)



This is not school, so doodling here at any time is okay, even encouraged. The word "doodle" signifies absentminded scrawling and is related to "dawdle," wasting time in inconsequential activity. Both words derive from the Low

Vome forth into thelight


Let Nature be your teacher. -Willian Wordsworth (177G-1850, Englishpoet)

ofthings, 3ff"*ffi:il.T1ff"J?;fffffi:f;:iHffi"#i:i aware of your doodles in a differenl way. psychologistsial doodles manifdstations of the unconscous. Letting our pencil flow without conscious intellectual control allows the archetypal pattems from within us to emerge.It is a geometer's method for self-discovery. But don't just take my word for this. Try everything yourself. Keep as your own only what your experience validates. As legal residents of the cosmos we have the authority to view the worid's blueprints. Here they are.Construct thi pattems that construct the universe.



Inr Mnnn r

hlhully 0ne You cannot conceive the many without the one. . . . The study of the unit is among those that lead the mind on and tum it to the vision of reality. . -Plato One principle must make the universe a single complex living creature, one from all. -PIotinus Everything an Indian does is in a circle, and that is becausethe power of the world always works in circles, and everything ties to be round. -Black Elk (1863-1950) The eye is the first circle, the horizon which it forms is the second: and throughout nature this primary figure is repeated without end. -Ralph WaldoEmerson(1803-1.882, and Americanessayist poet) It is.a fallacy of the old schoolsto divide man into parcels,elements, thoughts, emotions, intuitions, etc. AII human faculties consist of an intercormected whole. -Alfred Korzybski (1879-1950, American semanticist) AU are but pads of one stupendouswhole. -Alexander Pope I4/henceshall he have grief, how shall he be deluded who sees everywhere the Oneness? -Ishn Upanishnd I


THE CIRCLE DRAWSUS XII Benedictus wasselecting Pope century In thefourteenth artists to work for the Vatican,requestingfrom eachapplicant a sarnple of his ability. Although the Florentine Painter Giotto $24G1337) was known as a master of design and composition, he submitted only a circle draltm freehand, the famous "O of Giotto." Yet he was awarded the commission' Why? V[hat's so impressiveabout a simple circle?Cive a young child a crayon and paper and observe what he driws. At the earliest ageschildren scrawl lines and zigzags.Therecomesa time when they discoverthat a line's end can meet its beginning, and they take delight in the loop. an It continuesendlesslyaround and creates inside seParate an outside. Eventually, they come upon the circle. The from circle brings the loop to perfectiory so ror;nd in every direction. Children love to trace circular objects like cups and cans to achieve the fascinating perfection so hard for a young hand to draw ' . The discovery and appreciation of the circle is our early glimpse into the wholeness, unity, and divine order of the universe. Somepsychologistssay that the discovery of the circle arrives asthe child discovers the self and distinguishes himself from another. Even as adults our attention remains hypnotically drawn to circles, toward their centers, in objectswe createand thosewe see.We draw circlesand they oraw us. Looking at a circle is like looking into a mirror. We create and respondirresistibiy to circles,cylinders,and spheres of because recognizeourselvesin them. The message the we shapebypassesour consciousmental circuitry and speaks directly to the quiet intelligence of our deepest being. The circleis a reflectiono{ the world's-and our own-deeP Perfectiorg unity, design excellence, wholeness, and divine narure. To ancient mathematical philosophers, the circle symbolized the number one. They knew it as the source of all subsequent shapes, the womb in which all geometric pattems develop. The Greek term for the principles represented " by the circlewas Monad, from the rootmenein, tobe stable," and monas,ot "Oneness." In the alphanumeric correspon-


for Ancient Hindu priestsusedcircularmandalas practicein the mind. Although they havequite a differeni concentrating logos of purposein mind, modem graphicdesigners commercial that circlesirresistibly lure our attentiontoward their atsoknow center.

add up to dencesof the Greeks,the lettersof the word monas 361.The system allows a differenceof one, so the word for "oneness"becomes 360,not coincidentally,asthat is aiso the number of degreesaround a ful1 circle. Ancient mathematical philosophers referred to the Monad as The First The Seed,The Eisence, The Builder, The Foundation, The SpaceProducer and, most dramatic, The Immutable Truth and Destiny. It was also called Atlas because like the Titan upholding the heavensit supports, it corurectsand seParates the numbers it produces. The ancient philosophers conceived that the Monad breathes in the void and creates all subsequent numbers = Numbers (111111111 111111111 12345678987654321). x only express different qualities of the Monad. The ancients didn't consider unity to be a "number" but rather a Parent of numbers. They noted that unity exists in a1l things yet remains inapparent. They saw the relation of the Monad to a1lnumbers in a metaphor of simple arithmetic: any number when multiplied by unity remains itself (3 x 1 = 3). The same is true when unity divides into any number ( i = 5). Unity

(Greekfor The ouraboros " tatT-biter"), a cicular snake with its tail in its moutlL appears as a symbol on every continent representing eternity, not as all time but timelessness,


The Hindu deitY Shiva Nataraj,arguablYthe world's oldest continuously worshipped deity, represents the transforming life {orce that motivates all forms. Shiva is shown here dancing the universe into circular manifestation.

alwavs preservesthe identity of all it encounters'We might .uy ti-,ut"ott"" waits quietly within eachform without stir#g, motionless, nevlr mingling yet suPPorting all' The Moirad is the universe'scommon denominator'The ancient Gnostics called it the "silence force." The universe was carved of this primeval silence. Everything strives in one wav or another toward unity. There's more to a circle than iust a curved line' It's a wonderful first glyph of nature's alphabet. Every circle is identicai. Thev onlv differ in size. Each circle you seeor create is a profound statement about the transcendental nature of the uni-verse. Expanding from the "nowhere" of its dimensionless center to ihe infinitely many points of its circumference, a circle implies the mysterious generation from nothing to everything'. Its radiui and circumference are never both meaiurabl-e at the same time in similar rinits due to their mutual relation to the transcendental value known as "pi" = . 3.1,415926.. When either the radius or circumferenceis measurable in whole, rational units, the other is an endless, irrational decimal. Thus, a circle represents the limited and unlimited in one bodY. Our deepest awireness, the power that motivates all

In my endis my beginning. -Motto of Mary, Queen of Scots



The Tibetan \{heel of Life resemblesa Christian image of God holding the cosmos. (Engraving after the fresco by Piero di Puccio,c. 1400)


r It is Unitythatdoth me. enchnnt --4iordano Bruno (1548-1600, Italian .philosopher)



state," at Paolo)

of the World and the Expulsinn depicts the Gardery or "divine center of circular rings. Italian painter Giovaruri di

awareness,which we can call the "Power to Be Conscious," of which we are ngt ordinarily cognizant, recognizes its own transcendental nature in the geometry of the circle. For this {easonthe circle has been a r.rniversals}rmbol of an ideal perfection and divirie state that always exists around and within us whether we acknowledge it or not. Religious art has traditionally turned to the circle to symbolize this state of divinity ds "heaven " "paradise," "etemity," and "enlightenment. " Giotto's perfectly drawn circle communicated this universal ideal.

Beakice and Dante gaze into the concentric rings of light nearestGod.

TIPS ON USING THE IMPORTANT GEOMETER'S TOOLS Before we use the three classictools of the geometeDhere are some points you slrould consider: 1.. These ttuee tools---<ompass,shaightedge, and pencil-are a;cient, foundin various foimi in most


Lrrc uponline,lineupon a line;here little and there a little. -Bible (Isaiah28:10)

cultures. Used by artists, architects, and craftspeople, they areboth practical and symbolic' 2. Whether you use a metal compass or a string tied to a stick in the earth, these tools represent divine attributes. Treat them with respect. 3. Do nothing unconsciously. Be aware of each action you perform with them. No act in a geometric conitruction is trivial or without profound symbolism and correspondence to the world's creating Process. 4. Don't erasemistakes.Just as we cannot undo life's missteps,leave all marks where you rnake them and live with them until you can do the construction differentlY. Compass The Medieval geometerscontemplated the compassas an abstract s)'rnbol of the eye of God' h their worldview its legs represented rays of light and grace shining from heaven to earth, from deep within us outward toward the periphery of our ordinary awareness.The compasshas only one role: from a central seed-point it createsthe transcendental hole called the circle. It opens up a divine space of light, awareness, and potential configuration. Remember that every cir-


cle you construct represents the Monad, fhe complete unituming a circle, hold the compassat its top, not verie. \Atrhen avoid changing the size of the cilcle. Turn each cirits leg, to c1ein one complete rotation without hesitation, leaning the compassforward asyou tum. Avoid drawing only arcs and partial circles in the constructions. Complete all circles to see ihe full pattem of forces. Straightedge $Ihere circles intersect,points are created.A straightedge allows us to rule lines between points, revealing paths of energy, motion, and force between them. Your straightedge Measure is buili into the perforneedi no measure-marks. mance of the geometry The edge should be free of nicks. When drawing a line, put your pencil on one point and bring the straightedge against i! pivoting it to the second point. Then put the pencil on that second point to "test" its accuracyagainst the edge. Hold the straightedge firmly. Draw the line toward yourself in one steady stroke.Imagine the emerging line as a path of energy. Pencil The scribe system,whether pencil and PaPer or stick and earth, is the means by which an archetlpal pattern materializes.Pencil and paper translatedivine, eternal ideas into to accessible the geometer'ssight. \\4ren you coms1-mbols mit to creating a pencil stroke, move firmly, steadily, and deliberatelywithout hesitation. Keep the point sharp. Making a Point Creating one central point from which to radiate a circle is the beginning of al1 geometric constructiory no matter how The stepsinvolved in any construction complex it becomes. are a metaphor for the stagesof the divine, ongoing creating processitself. Anywhere you piace a centeryou can scribea circleand spnbolically createthe spaceof the universeitself. The circle is a shape begging to be organized. As the value "one," or unit, is the parent of all numbers, the circle

art,Iad, the as True Thou to its circumference center. -Herman Melville (1819-1891, American novelist)



The circle accommodates all shapes within itself.

ard The wortdis single it into came being from the outwards. center -Joannes Stobaeus(fifth century .t.o., Greek anthologist)

o@oo is the parent of all shapes.All subsequentshapesand patMonad' tems will be inscribedwithin this all-encompassing (Latin for "one That's why the world is called a universe tum"). No other universes exist excePt within this One, which Plato refers to as the "whole of wholes."

o To construct a symbolic universe, begin with a poin! the Hold the closed cornpassupright, and poke circle's essence. a s1'rnbolicpoint upon the paper. A true point is impossible to draw, having absolutely no dimensiory not length, width, or height. Ours is not a tuue mathematical point because seen r-rnderthe microscope it's a three-dimensional heap of carbon. Ideal geometry is impossible to perform with human tools. We can only symboUze ideals by the geometric shapeswe construct. The closed upright compass on the point represents the mythological world axis, world mountain, or holy center of many cultures, the s1'rnbolic pole or spine that supports creation and around which everything turns in adoration. Traditionally, the center is the most honorable place, knovvn to the Greeks as "the keep of Ze:us." Protector of hearths and boundaries (centersand circumJerences)and the source of moral order, Zeus dispensed judgment from the center. Nothing exists without a center around which it revolves, whether the nucleus of an atom, the heart of our body, hearth of the home, capital of a natiory sun in the solar system, or black hole at the core of a galaxy. \Alhen the center does not hold, the entire affair collapses. An idea or conversation is considered "pointless" not because it leads nowhere but becauseit has no cenierholding it together. The ooint is the source of our whole of wholes. It is beyond understanding, unknowable, silently self-enfolded. But like a seed,a point will expand to fulfill itself as a circle,

himself ltod makes known to the world; He

circle of fills up the zahole but his theunit:erse, makes in particularabode the whichis thesoulof centeL thejust. -Lrucia]r. (c. 24c_312) , Christian theologian)


Geometric construction can be used as a form of meditation. Ponder the point as a seed enfolding a sacred mystery. Hold that sense of wonder through the full construcnon. We can use this approach to symbolic geometry as sacred geometry by experiencing what the construction process symbolizes within us. We can do this by shifting attention from outside to inside, from the page to its viewer. Consider the point as symbolizing your own " cenIer." What center? Dancers and g)'mnasts gracefully work with the body's center of gravity to balance during motion. Undersea divers walk around at the bottom wearing a twenty-pound weight at their groin, the center of gravity. Even when sitting motionlesseveryonehas a psychological center of gravity, the thoughts, emotions, or desires with which we identify and from which we view the world at any given moment. Through meditation and self-contemplation we can seek a more subtle center, our higher or deeper self, thepoTrerthat motiztates acl:.ons, tine emotions, thoughts, and desires. This center of gravity is not in space but in pure awareness, the "p1ace" in you now aware of these very words. V/here do you "hear" thesewords? You may believe or doubt that there is such a center,but vou cannot doubt the powet wlthwhich you doubt. This power is ihe. motivating power-with-which-we-are-conscious, identical to the heart of every natural form and syrnbolized by the center of the circle. It is the seed of our mvsterious abilitv to be aware. OnJy during psychologicalstillness,with no mental, emotional, or desire ripples disturbing the quiet "pond" o{ awarenesswithin us, can we consciously approach our own deep sacred center. \44ren you mark a point on paper, contemplate it as representing a center of profound awareness about to expand, construct, and motivate the universe. Then look inward, silence the voices, and seek that center's correspondencewithin you.


setet orc Godhead,

allin all beings, pelI)dding,the inner SeIfof all, presiding oaeraII action,witness, conscious knowerandabsolute. . . theOnein controloaerthe Tnany whoarepassiae to Nature, one fashions seed in manyways. -Swetaswatara Upanishad

As on thesmooth expanse crystal of lakes, thesinking stone first a at circle makes;

TO THE MARGINDANGE The point is rnade, the center established. The compass standsupright on it. Now we open the compass. This seemingly trivial act is an important stage in the geometric

Thetrembling surface by themotionstirr'd, Spread a second in circle, then a third; Wide,and morewide, the anc floating rings ada e,

FilI all thewat'ryplain, and to themargindance. -Alexander Pope



the [t rePresents metaphor of the cosmic creating process' expanslonm first irchetypal principle of the Monad: equal -all directions. mvths, the universal creating processbegins ;;";; very nrst *ith an expansion from a divine center' like the Light"' In Hindu {Ythol"Let there be ;ili;'";;ft, Brahmaspeaksaloud the wo'1.ono^: the dimensionless oev, lefiersor "i'A^," u word made of the first, middle' and tmal. which rePresentsthe circle's tfu-ee ttt"^-d*tUii "fphabet, parts; the centei the radius, and the circumference' and our and ti':: iwn spiritual center,psychologicalreaches' 11? rePresenF th9.,hrslm11: rial f&m. The opening comPass festation of God's light and Brahma's voice' iltummaflng

A point within a circle was the Egyptian, Chinese, and Mayan glyph {or "light."

Riooles expanding in a pond presentan optical illusion' We we ur!o-" thut thu *"uter ii moving outward but it's not What racing outward seeis a wave of erergy from the irnpacting pebble through the water equally in all directions Watch a floating 'ihe objectoniy bobs up and down while obiecLs waves passby. horizontally.Visible mafler merely makesthe the wave traverses energy pattern visible. ln a small puddle, attractionamong the wateimolecuies holds the wave together' Large ocear waves are held together by gravitY.



and vibrating the universe into existence, as expanding states of self-awareness,which we call "nature." Natue's forms represent invisible forces made visible. The force of the circie's equal expansion works through different materials. Tap the side of a round cup of liquid, and watch as perfect concentric rings appear and converge to the center, therr pass it and expand outward again. Nature delights in the principle of equal expansion in concentric ripples, splashes, craters, bubbles, flowers, and exploding stars. As you open your compass,consider that you are metaphoricaliy repeating this first principle of the Mona4 the opening of light, spacg time, and power in all directions.

A Greek amphitheater's circular design matches the shape of the invisible but expanding sound waves of the actors' voices and provides everyone with an equal view.

Impact ("spiash") crater of milk, like a crater on the moory is a diagram of expanding forces. The energy spreadsas ripples on a pond but is limited by the matedal. The splash of every sphereskiving raindrop speaks the principle of the Monad.


is not thewater.

Thewatermerely us told about.the fiooingby. waae -R. Buckminster Fu-11er (1895-1983,American inventoD engineer, poet mathematician, and futurist)


,r wooD


Tree stump. Since plants are composed mostly of water (organized by minerals), it's natual that their growth rings resembie watery ripples expanding very slowly acrossa pond.



T}IE WHEEL WHIRUD Beholdthe world, how it is whkled round, And for it is sowhirfd is namedso' juristandpoet) (1569-1626 -Sir lohnDaaies , English The secondprinciple of the Monad is expressedby the circle'srotary motio;. Unlike the still center,the circumference speaksof ino.rument.We replicatethis universalprinciple in Jur geometricconstructionswheneverwe tum the comPass in uro,-ild it, point and scribe a circle' S)rynbolized nearly nature's unieverv crrlh.lieas a wheel, the circle represents .r.rsil cvcles, circulations, circuits, orbits, periodicities' vibrations, and rhYthms. cyclesare a principle of the Monad, they areallBecause oervasive in lhe r.rniverie.We are thoroughly enmeshedin ivcles and periodic rhythms but notice only the most obvious, like our breath and hunger or the time or season'


Cycles characterize biological and inorganic processesin all organisms from simpiest to most complex. The life cYcles of plants and animals, from seed to fruit, all migrations, metamorphoses, hibemations and reproductive cYclesare qmchronized with seasonal periodicities.


#t;r. ui&"r\a*rr wrJiletc^tto',tf'r


15 The Rock Cycle. Considerthe slowly changing rocks that mesh their own long cycles of weathering, melting, cooling, crushing, compression, and reformation within a greater whole. The global circulation of weather, as well as the planetary water, oxygen, and nitrogen cyclesare also well known.



lo flEto ttlRr E! tult S 9 EIN

tlrt IoLl


x r 0 f iE l 3 PtttaitE! lE03

Circulatory System. Our heart is at the "center" of con-



one All cvcleshave rising and declining phases'When is true on any scate' Ezten the seasons a side goei up, the other goesdown' This form i" ,ui"itrg;,}'""Is and ii the rising/falling.PYl":.of -:T: greatcirclein their tional buist, the changing amount of daylight'ht?ign ll" and changing, alwaYs tne rre the seasons, rise and decline of great cultures' and againto where cvclesof stars. back come -' theywere.Thelife of a Obr"*" a rapidly cycling bicycle wheel,or ceiling fan' When it revolveJ slowly we can seeeach individual spoke manis in a circle from it tulns faster our nervous system iust o o childho d t o childho d and or blade. But when register the revolutions as discontinuous' and where carmot f1" soit is in eaerything 6evo"a aU"o"t1,550cycles per second the spokes""d our Tnoaes. teepees ap'pearasa solid disk. Look around at any "solid" obiect'Be poTt)er of of a#are that the appearance its "surface" is due to raptdty wereroundlikethenests atoms,which move so fast asto give our nervous oscillating were birds,and these

alwayssetin a circle,the nation'shooP. -Black Elk

Illusion of wheels. Observe the spokes of a bicycle wheel as it spins. The ends near the rim are rapid and blurry while the ends near the hub move slowly and are clearly seen.The paradox of the wheel is that the rim ictually does spin faster than its hub because,while they turn acrossthe same angle, the rim must travel a lonqer distance in the same time. To do so it moves faster. WheJls, gears, cranks, dials, knobs, levers, belts, and ball bearings use this property for magnifying, diminishinC'Td trarsfe"rring mechlniial power. For instance, a revolving door is easier to push toward its outside than near the center pivot'



system the impresiion of a smooth surface. The same is tme for our senseof hearing. A card held against tuming bicycle spokes or the teeth of a turning gear will produce disclete sounds until the rapidity is such that the sound is perceived as a continuous hum. A1l sensesare fooled by rapid cyclic vibration so that even texture, smell, and taste aPpear as continuous to our registration faculties. Every process is characterizedby cycles. The appearancesof the entire world with all its natural and technological cycles are images rooted in the archetypal cyclic principle of the Monad, tepresented by the geometel/s tuming compass. Cooperating with nature requires that we recognize the existence, and learn the ways, of its omnipresent cycles.

the fVot ,o arresting aast of wheel Time, still and Thatround round might. turnswith onward -Charies C. Clarke (L787-1877, F,nglish critic and scholar)


in career begun hod

and Washington it would end there.I liked the ideaof a circle beingcompleted. -Helen Hayes (190V1993, American actress)

Ashonomical cycles.The nestedcyclesof the moons,planets, suns, and galaies are documented by astronomers. Moons orbit planets which revolve around suns. Spinning star clusters group as revolving galades, which whirl among other spinning galactic dusters, parts of cycles beyond comprehension. Here, as sem through time, the stars seem to trace concentric circles around the Pole Star.

is Ihe sage hewhohas attained centralpoint the it of the wheeland pro?re6 without himself in participating the moaement remains and boundto the Unaarying Mean. -Tr^ier crr'ino



Cyclic Time lnstruments of time reckoning from sundials and calendars to atomic clocks are structural sr..rnbolsof astronomical, atomic, and biological the cvcles.Hands of the analog (round) clock model atoarent path of the sun through the sky,or the f]arth's dailv spin seenfrom abovethe South lole' Round Az-teccalendars (and those of other cultures) were arranged on the ground as models of their cyclic calendar, socially and symbolically meshins th; shucture of time with that of spacein rounds oficricultural rituals, fairs, and festivals around a circie of locations through the year.

Asar (in Greek and English Osiris), the Eglptiar personificationof cycles.ln general,gods and godof desses the ancientworld were personified archeh'pal principles whoseactionsmanifestthemselves (symbof in nature.Aiar, o{ten depicted green-faced personifies all cycles in nature. izing vegetation), Here he revolves in the "waters" of chaos, lifting the sky goddess Nut into spiritual light.

4 G

.} \



PERFECT SPACE FOR A UNIVERSE A circle may be small yet it may be as mathematicallybeautiful and perfect as a large one. -Isaic D'Israeli(1766-7848, Englishmanof letters) The third

has The wheel come full circle. -William Shakespeare

A the areawithi| tlre circumference. circleis not just the miraculous spaceinside, which manifests curve but the


involves PdnciPle of the Monad


17 A sirnple experirnent with a single loop of string and graph paper, chessboard,or checkered tablecloth will prove the supremacy of a circle. Form the loop into different shapesand lay eachover the squares of the graph paper. Count the number of squareseachshapeencloses. Although each shape has the identical perimeter because the loop's length doesn't change only the circle covers the most squares.

between nothingness (zero-dimensional point) and everything (infinitely many points around the iircumference). A circle expressesthe most practical and efficient geometric spacefor natural and human creations to occur. Of all shapes, a circle encloses the most space by the smallest perimeter.In other words, themostenciosure with leastexposure.The Monad's third principle is maximized efficiency. Look for circles, cylinders, and spheres and you,il see them ever).'where: A round shield gave the ancient soldier maximum protection behind the largest area while employing the least material and having the least weight. Manholecoaers. Did you ever wonder why a rnar*role cover is round? A circle is the onlv shar:e that won't fall into its own hole. A "qua.e or triangular cover could drop in along its diagonal, which is longer than any side, but a circle'sdiameter is everywhere the same. Ring roads. Ring roads or beltways around every major city and town allow a traveler easiestaccessinto the center of a city Shields.

The rnoving path of a fixed point on a rolling circle leaves an arched kail. Maintaining unity, the two shaded portions on each side of the circle have the exact area as the rolling circie that produced them. Verify this by taping a pencil to a metal can and rolling it along the bottom of a wall at a right angle io graph paper on the floor





When a wagon train formed a circle for protection it enclosed the most ground while offering the least exposure to attack from outside.

from anywhere on the periphery. A city, a cell, and a nation are considered independent when they control their own borders.

and Cups cans. A cylindrical cup or can containsmore material within it ttun a square- or hexagonal-shaped container, made of the sameamount of material, does.The circular shapeallows it to hold more using the least material and weight. Only a sphere would be more efficient ftut impractical to stack). A round dinner plate holds more food than other shapei while allowing the least opportunity for it to fall over the side. A round pizza usesthe leastamount of dough to fill the largest spaceand provide the greatestareafor toppings.

not Cirrtrc areprais'd tha! abound largeness, In round:So but th' exactly thnt doth Iifewe praise, Not ex.cel in muchtime, Plates. but actingweII. -Edmund Waller (1606-1682English poet)




A round table is an expression of the equality of the diners. Its maximized sutface area allows them to share more dishes than any other shape with the sameperimeter would.

Bird's nest/Igloo/African hut. Animal and human builders in all climates prefer cicular and hemispherical homes that enclose the largest living spacewith the minirnum exposure while using the least materials, time, and energy to build.



thenhowlofty Consider wide and hoTD

of is theexcellencethe Worth Eternal in ushich somanyrnirrors candioide and its power maiestY , foreaermore as Itselfremaining One,as It wasbefore. -Dante (126F1321)

Exile is tenibleto those who hnoe,as it were,a ed circumscrib habitation ; wholook but not to those globeas uponthewhole onecity. -Cicero

expressedas a point and a-cirThe Monad, or oneness, cle, is the foundation for our geometricconstructionot trre universe. The three parts of the circle-center, circumterence,and radius forming the spacewithin-correspond to cycles, the tfuee principles of the Monad: equal expansion,. ipace. These principles, along with ihe and efficient Monad's wholeness, are all-pervasive and lie at the {oundation of the world's obiects and events, as the number one is hidden within every integer. The Monad is knowable to us through its expression in nature's designs and human affairs as equal expansion, cycles,and efficient space.Natural structures are universally recognized as beautiful and most efficient. We, too, are Part of the world's harmonious design and can't help but express the Monad's principles in the things we do and create. Everything seeksunity. The goal of many religions and mythic ordeals is to return to a lost state of Divine Oneness. because But we have no needto teturn to a stateof oneness alreadyare integrated in it. Barely unity is axiomati c and we recognizing our situation, here and now we live in a whole and beautifully harmonious wonder world. Only a selfkeepsus from recognizing imposed illusion of separateness of awarenessand identity with the One. To out own center understand this unity the ancient mathematical philosophers contemplated the principles of the Monad through the irithmetic principles of the "number" one and by exploring its geometric exPressionas the circle. is This illusion of separateness not a direct characteristic of the Monad but requires twoness' The Two, an "other," proceeds from the One for the process of universal conitruction. To understand how duality and the senseof separation derive from unity and work in the wo d and ourselves, we must go deeper into the mythic realm of geometry and number to the principle of twoness, known to 'the Greeks as the Dyad.



ItTrkesTmh The opposite is beneficial; from things that differ comes the fairest attunement; all things are bom through strife. -Heraclitus (c. 540-+.480 t.c., Greek philosopher) In the Two we experience the very essenceof mrmber more intensely than in other numbers, that essence being to bind many together into one, to equate plurality and unity. Our mind divides the world into heaven and earth, day and night, light and darkness, right and left, man and woman, I and you-and the more strongly we sensethe separation between these poles, whatever they may be, the more powerfully do we also sense their unity. *KarI Mmninger (1893-1990, Americanpsychiatrist) This whole wide world is only he and she. -Sri Aurobindo Ghose


Birth of the Line. 2l



THE BIRTHOF THE OTHER The circle is the womb and cradle of our s).rnbolic universe. It shows us the characteristics of unity, everywhere the same and containing no differences within it. Like a pebble tossed into a pond, a circle can only reproduce more circles in its own likeness. The ancient rnathematical philosophers saw this in the metaphor of arithmetic. They noticed that no matter how many times unity is multiplied by itself, the result is the same: one(1 x 1x L x... x 1x 1= 1).Sohow does unity, oneness,step beyond itseH and become the many? How can the Monad generate the other principles, other shapes,other numbers? How does the "same" produce an "other" How does the primeval "I" generateits "Thou"? With a mirror. It simply needs another circle identical to itself. The circle replicates a mate for itself by contemplating itself, reflecting its light, and casting its own shadow. The processis mirrored through geometry as the birth of the line. It has been done from time immemorial through an ancient geometric construction called in Western tradition piscis (Latrn for "bladder of the fish"). Try this: the aesica (1) First mark a center and spin a circle with the compass. (2) Lift the comp ass .without changingthesizeof its openingand reverse it. Place its sharp point anywhere on the circumference to make a new center and swing another circle. This

.fl\ A line is bom within the oesicapiscis.





second center is actually an emanation of the original circle since it resideson its circumference.(3) Theseintersecting circles,,linked across their centers to form a line, make an ancient afid obvious symbol of twoness.The overlapping piscis. spacebetween them is the aesica As if across a mirror, the archetypal line is drawn between the two centers. Sit up straight and get a senseof your spine as a line. To a symbolic geometer a line is a picture of energy, tension, force, actior1 impulse, urge, directiory movement. The straight lines we wiII draw in our constructions represent the tension and motion between the poles of every creating l]rocess. A "true" archetypal line is one-dimensional, having length but no height or thickness. It simply expressesthe principle of extension.Like a "true" point a "true" line is impossible to draw with material tools. But these two, the point/circle and the 1ine, are the parents of all subsequent geometric constructions that do manifest themselves in actual nafure.

Phantom circle. With your face a comJortable distance from this circle, relax your eyes and let your focus drift until the circle seemsto separate into two side-by-side circles. When they seem to touch each other's center, adjust and hold the focus. This illusion is the result of one awarenessviewing through two separate eyes.


t-u Rumor dothdouble, Iiketheaoice echo, and thenumbers the of feared. -William Shakespeare



plscisto divide a line in haU. Using lhe aesica

THE DUAL THRONG The principle of "twoness," or " otherness,"was calted Dyad by the Greek philosophers of the five centuries before Christ. They were suspiciousof it because seemedto revolt from it unity, distancing itself from the divine Monad. They refened to the Dyad as "audacity" for its boldness in implying a separation from the original wholeness and "anguish" due to its



Coupl* arcwholes and not wholes, whatagrees disagrees, concordant the is discordant. Fromall things oneandfrom oneaII things. -Heraclitus

Loop: two separatesides.

Mdbius strip: one-sided. Mdbius strip: two as one. Give a strip of paper a half twist and tape the ends together. This Mcibius strip has only one side and one edge,although it appearsto be two-sided. Tiace your finger around the face and you'll return to the same beginning. It's a thee-dimen, sional symbol of the Dyad, of one appearingas two.

inevitableyeaming to retum to unity. It was alsocalled "distress," "falling short," "the 1ie," and "i11usion"since they believed the Monad alone was all. Today,we employ this negativeaspectin the derogatory phrases "two-faced"and "speaking with a forked tongue." The principle of the Dyad is polarity. Polar tension occursin all natural and human affairsasany opposing relationship, contrast,difference.It is at the root of our pernicious notion of separateness from each other, from nature, and from our own inherent divinity. The paradox of the Dyad is that while it appearsto separate from unity, its opposite poles remember their source and attract eachother in an attempt to merge and return to that state of unity. The Dyad simultaneously divides and unites, repelsand attracts,separates from unity and craves to return to it. A line createsboth a boundary that divides and a link that binds. We know we are under the sway of the Dyad when we are attractedor repelledby anything. The linguistic roots of the word "two" support its dual r.atluole. Separation emphasizedin words having the prefix is " di! or "du-": discord, difference, dispute, dissen! disunion, difficult, dilemma, divide, distinct, distance,distinguish, disgruntled, distribute, diverge, dichotomy, doub! duo, deuce,duet, dual and duel. Through linguistic evolution this root also gave us the words of personal polarity "thou" and "yon." Our sad and dangerousnotion of separation from nature and eachother is rooted in the principles of the Dyad. On the other hand, another twoness indicator, "tw-," l.rnpliesjoining, as in two, twin, twain, between, twine (two twisted threads), and twilight (a blend of day and night). There seem to be more words emphasi zing " sepatation" than "joining," implying an alienation from unity. Polar tension is at the root of all birth and creation.As the popular phrasereminds us, it takestwo to tango.Exactly two people of oppositegender,no more or 1ess, produce can a child. \ /hen cool, dry air penetrateswarm, wet air, rain precipitates.Woven cloth manifestsitself at the intersection of warp and woof. Two poles of a battery,positive and negative, are needed to complete an eiectric circuit. Two fixed



ends of a guitar string allow us to pluck it, creating vibration, sound, and music. One chopstick is motionless, the other moves, and together they can pick up food. There isn't anything composed of matter (or antimatter) that avoids polarify. Even the geometric compass operates by the interplay of two legs, one motionless and the other moving, the poles of centerand circumference. It The Dyad is the basisof every creativeprocess. shows up as rhythmic oscillation between opposite poles, as close pulsing at the edge asour beatingheart and asfar as quasars of ihe universe. Human nature mirrors outer nature. All personal relationships have at their essencethe'archetypal tension between opposites. Polarities drive the dancesof mating, love and hate, taking responsibility or assigning blame, they are integral to political oppostrength and tenderness; sition parties, diplomacy, businesspartnerships and business rivalries. Psychologiststell us that within each of us are the poles of anima and animus, female and rnale aspectswhose relative balance determines how we relate to other men and women. Although we think we act independently, we {ollow nature's polar principles in most everyihing we do. A look at language reveals our investment in polarization. How often do we hear the words good/evil, true/false, hot/cold, aiways/never, win/lose, pass/fail, input / output, us/them, this/that, right/wrong, high/low, physics/metaphysics, man/nature, contract/expand, constructive/ desfructive, visible/invisible, over/under, body/mind, thesis/antithesis , rtp / down, Iatgh/ cry, permanence/ change, virtue/vice, presence/afsence,pressure/re1ease,. joy / despafu, health/disease,too litile/too much, success,/ failure, creature Creator, pleasure/pain, profit/loss, lim/ ited/unlimited. our private world i; filled with thesetwin Iabels and the pictures and values that accompany them. This is the great "dual throng" of language. Every entity confrontsitself with its own opposite.For eachTWeedledum there must be a Tweedledee. Neither word in anv r:air can exist alone.Eachimplies the other.If there'ssomettLing you don't like, you can assume that its opposite exists, which

shouting Errryboay', "IMich sideareyou on?" -Bob Dylan(1941)

in Thrrc i, noquality this world that is not whnt it is merclyby contrast. Nothing existsin itself. -Herman Melville

tEaerything that ^!;o;hn+ot h^n the tree nf originatedfrom +ha+oo o1 knowledge carries in it duality. -Zohnr (mystrcal Jewishtext)


t Jaery Language naotng a


structure, the by aery nature language of , rcflects in its own structurethat of theworld asassumed by thosewho eaohted the In language. otherwords, we readunconsciously into theworld thestructureof thelanguage use. we -AJfred Korzybski

you will like. But shunning one while chasing its oPPosite only invokes and strengthens the one we try to avoid. Accepting one gives equal strength to its opposite. The words we speak, write, and think will unconsciously help determine what we see;thus, our language can shepherd us into defending ideas we may not actually hold. But experience doesn't support two-valued orientation. Life-facts are in between.That's why the ancientscalied Dyad "illusion." Under the sway of the Dyad we seewalls, boundaries, dividers. Polarized thjnking encourages oul senseof separateness and deflectsour vision from the world's-and our own-inherent unity. We cannot live in a senseof oneness, uniry wholeness, completeness while bouncing in the mirror worid of implied opposites, continually attracted or repelled, feeling separate from each other, from nature and from a deep relation to our inner selves.This notion of separatenessfrom nature, in fact, encourages the environmental crises we find ourselves in. We would not attempt to tame, master,plundel and foul nature if we recognized how those acts affect ourselves.To avoid disasterwe must leam to think differenily and avoid the traps of polarity in everyday language. Even the most seemingly irnocuous statement blinds us to what's real1ygoing on. For exarnple,when we say, "T?risworld is not real; reality is somewhereelse," or "Heaven is my home, not this world," or "Ideal geometric shapes exist in an archetypal realm, beyond our knowirirg,"we're ensuring that we won't find what we seek.By thinking thai what we seek lies elsewhere, we miss seeing it here right before us. Ideals aren't elsewherebut are embedded in the event.We don't notice because we're lured to one pole or the other. Only by acknowledging both poles in the pair asinseparable we overcomerelative duality and get can to their common sourcein the Monad.In our deepestSelfwe are beyond all polarity.

IT'SA CI{ECKERBOARD WORLD Many worldwide religions and mythic cosmologies include storiesof the birth of a primeval dichotomy,usually lovers, enemies, or twins, whose polar interplay generates the world. The myths are personifications in ancient terminol-



Syrnbols of the primeval polarities of light and dark that interact to generate the world include the Taoist yin-yang, the Mayan Hunab Ku, and the Mayan glyph signi$ing "eclipse." Each has a bit of its opposite within itself that drives it endlessly around in search of its complement. The chessboard's alternating dark and light squaresrepresentthe field of the world. The Navajo creation myih originates with a primal mother and father, earth and sky. A Hindu story of creation involves an altemation of pulling on opposite ends of a serpent by deaas angels") and asuras(" demons"), (" chuming cosmic milk to generate the worid. An Egyptian symbol of thg nation's uniry called sn4d larl, depicts the antagonists Set and Horus pulling on papyrus and lotus flowers, symbols of No{th and South Egypt which grow at oppositeends of the Nile.



ogy of primal polar principles known to modern scientific understanding. Through these stories, the dance of opposites in natule is mirrored in ourselves.Thesesymbols and the works of art employing them are part of larger myths that describeways to passconstructivelythrough the polarized world.

LOVE AMONGTHE ATOMS Scienceand technology cannot avoid encountering the everpresent principie of polarity. It is so basic that perhaps insteadof teachingscience youngstersin separate to pigeonholes of biology, chemistry, physics, and so on, science courses could investigate the principles that run through eachof them, such as wholeness,polarity, balance,pattern, and harmony. The Dyad's fundamental characteristicis the existence of a pair of distinct but equal oppositesthat seekto unite in an urge to return to unity. Modem scientists know this characteristicasthe "laws" of attractionand repulsion.From this basic polarity, light, energy, and matter eventuate. For anything to manifest there must be polarity.

For eoery actionthereis an equnlbut opposite reaction. -Sir IsaacNewton (L642-1727English , mathematicianand physicist, his secondlaw of thermodynamics)

Electronorbit around a hydrogen nucleusresembles a planetary orbit around the sun.







The Door of Hgdrogen Like a door from the Monad, the fust atom of the ninety-two natural elements to emerge is hydrogen. It is a pure representation of the Dyad, the only element consisting of only two components: a positively charged proton at its center and a negatively charged electron orbiting it as a circumference, A1l other atoms have a third neutral component (see chapter three). hr the role of.the Dyad, hydrogen is a door between the ur*nown unity (Monad) from which it emerged and the subsequent elements ("the Many"), which are built by fusing hydrogen atoms together. "Stotic Cting" Sinceelechons occur in materials in large numbers, some of them can be loosened by mere friction and induced to move. ScuffLingone's feet acrossa carpet on a dry day loosenselectrons, which run up and sit on the surface of the skin. They jump at the first opportunity to unite with an oppositely charged "iover" such as a metal doorknob. The spark is sim. ply the result of a primal urge of parts to retum to the whole, of polarity's craving to merge to unity, the original state of the Monad. Even the everyday "static cling" of clothing and plastic wrap illustrates the urge of the Many to retum to the state of the One. Scientists call the strength of the "1ove" between poles ther:r"voltage."

Magnetic field. Magnets are pure expressions of the Dyad. Experiments with them demonstrate its principles: opposite poles athact each other, and alike poles repel. No magnet with other than two poles has ever been found or can be created.



Ce1l division. The human body grows through the process of mitosis, or cell division, as each cell replicates itself and then divides into two identical "sister" ce1ls.The Monad becomes the Dyad as one becomes two, which becomes four, eight, sixteery and so on r.ntil we have a complete cellular organism: the Many.

@@@@@ THE BIRTH PORTAL The study of number-words in many cultures reveals an early awareness that multiplicity in nature and arithmetic comes from the blending of opposites. For example, the ancient Sumerianwords for one and two are also those for man and woman. Today, we consider one and two as merely number quantities, not realizing they are s).'rnbols basic facts of of existence. Surprisingly, ancient mathematicalphilosophers did not consider one and two to be numbers thernselves since their representations-point and line-are not actual. A point has no dimension and a line just one dirnension. Nobody can hold a true point or line in his hand. Likewise, no one or two points, lines, or angleswill createany acfual forrn by themselves. But an ongoing interplay begiming with a point and line is all that's required to construct the world's geometric pattems. Thus, the Monad and Dyad were consideredby the ancientsto be not numbers but the parentsof nrmbers. Their mating, the fusion of the principles of one and fwo, point and 1ine, unity and difference gives birth to all subsequent archet1pal principles revealed as numbers, symbolized by numerals, and seen as shapesin nature. The Dyad, then, is the doorway between the One and the Many. The ancient mathematicalphilosophersdiscoveredthis by studying arithmetic. They noticed how numbers act

1+1>1x1-TheUnique 2+2 = 2 x2---TheDoor

3+3<3x31 4 + 4 < 4x4 |The Many 5+5<5x5J The simplest arithmetic properties of any number reveal its deepestnature. Two is the balance or door linking the One with the Many.



For when added to and multiplied by themselves. example, "one" is the only value that, when added to itself, produces arcstlt greaterlhan when it is multiplied by itself. One plus one is morethan one times one. "Two" is also unique but in a different way. This is the only casewhere the addition of a number to itself yieids the same result as it does multiplying by itself. TWo plus two equals two times two. Two represents a balancing point between unity and all subsequent numbers, between One and the Many. The remaining numbers three, four, five, and so on have in common the fact that when each is added to itself the result will always be lessthan when each is multiplied by itself. Three r:1usthree is lessthnn three times three. The same for four, fiv4 six, and forever onward: the Many. Symboiically, "two" acts as an intermediary a transition, a door or portal between the Monad and all the rest of the numbers..TWonessis the hole or lens through which unity becomes and balanceswith the Many. This is the geometric lesson of the two linked circles, symbol of the Dyad. The almond-shapedzone of interpenetration between the circles has attracted the attention of geometers, artists,architects, and mythmakers through hispiscis,inChristian cultures a reference tory. This is the oesica to Christ as the "fish" in the Age of Pisces.It's called a mando a (" alnond" ) in India. It was known in the early civilizations of Mesopotamia, Africa, Asia, and elsewhere.

piscis,or mandorla Vesica

Pairs of "guard" cells on the underside of a leaf contract and relax to form a aesica piscls-like opening called a "stoma," allowing the plant to exchange oxygen for carbon dioxide and breathe.



This almond spaceis the crucible of the creating process. It is an opening to the womb from which geometric forms are bom. It brings forth shapesand patterns from the arche, t1pa1world of ideal geometry. Th e oesica piscisis a yoni (Sanskrit term for the femaie generative organs) through which the geometric shapes and pattems of our universe emerge. For this reasonit has also been called the womb of Chaos, the wornb of the Goddess of Night, and the mouth that speaks the word of creation.The z;esica plscisis where the geometerassistsin the birthing process.

Corporate logos are often designed around the uesica piscis.

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n \1 Activity: Doodling with ztesicas pisces. \tVhat is often called "doodling" can be of great value to the irnaginaiion of a geometer. Iny'hen one turns off the mind,s logical mode and doodles intuitively, discoveries cal take place that illuminate basicgeomefricrelationships. doodle within So and around these intersecting circles. Some have been started. Connect points, follow curves, shade areas.Doodle however you like.



Discoverthe YesicoPiscis in AncientArt ond Architecture was used functionally and symbolically in piscis aesica Tlne the constructionof doors or portals betweenmundaneand these toolsto replicate Use spiritualspaces. the geometer's styles, representingcreedsthat are different but different sharethe sameSeometry.

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is and Etemity; theret'ore, of theonlythingincaPable in and modes fashions its

principles. -Sir ChristopherWren (L632:17 Englisharchitect) 23,


The Moslem arch and cathedral doorway are both designed using tlvo intersectpiscis, ing circles and oesica but at right angles to each other. Through these doors, symbols of spiritual passage, we leave the street of the Many and enter the dornain of the One.

The west facade of the Cathedral of Notre Dame in Paris is designed as a circle descending from heaven above which intersects a circle rising from earth below. Entering a cathedral symbolically places us in the spacebetween them where transformation is possible.



Hindu temples are traditionally designedupon geometry originating with the zesica plscrs. Templesrepresenta portal between eafh and the heavens,our mundane and spiritual identities. The geometry within the circles determines the floor plan of the temple, revealing the detail to which they were planned.

Design using the zesicapiscisis sobasic that we find it underlying human creations thousands of years apart, as seenin this jeweky of the Egyptian pharaoh Tut-Ankh-Amon and the British RoyalSeal. Lntersecting circles havebeendrawn within them.To seehow their designs stem from the geometry, draw staight lines between the intersections of the circles and their centers, extending them to the opposite sides of the ztesiu piscis.

The Birth of the Mong One pebble tossed into a quiet pond imparts energy we can see as expanding concentric rings. TWo pebbles tossed simultaneously a distance apart create intersecting rings and weave a mathematical wonder: concentric, w atirv aeiicas pisces.Nodes or points occur everywhere the circles



cross.Remarkably, these points establish the precise mathematical distances and relationships required to construct the basic geomehic shapesand pattems that recur through the universe.Because circlesare similar to eachother regardall less of size, the geomehic relationships remain constant no matter how large the ripples become. Sincethe Pythagoreans considered the first ten numbers to be seed-pattemsfor a1l the principles of the cosmos,a geometerneed only createtheir shapesto model aII the universal rhythms. The first three shapes to emerge from the aesicapiscis, the triangle, square, and pentagon, form the only relationships or ratios required to generate all the rest (except for sevenness: see chapter seven). These relationships,called the squareroots of two, three,and five,looklike numbers and are callednumbers, but they aren't like other numbers. They are like seedsfrom which the shapesemerge and are nourished. These "root" relationships are expressible not as whole numbers but as never-ending decimals (1..41.42135.; 1..7320508 .; 2.2360679 ...). As statements .. .. . rnade by the geometry of two intersecting circles and the aesica piscis,these ongoing relationships hold the structural patterns for all numbers and shapesthat fol1ow. The "parents" of numbers, symbolized by Monad and Dyad, one and two, point and line, when united as a oesim prscls generatethe mathematical"root relationships"underIying the shapesand patterns of the natural world. I,Vhenthe distance between centers is one (unit) then the relationships reveal themselves.They're responsiblefor the basic twb-

Oh, gront myprayer, *, that I may neaerlose the touchof theonein the play of themany. *Rabindranath Thgore (1861-1941, Indian poet)



dimensional shapes, the triangle, square, and Pentagon. When conshucted on paper the shapes may be folded to make the five equal divisions of three-dimensional space known as the "Platonic volumes." Each of these shapeswill be constructed and explored in subsequentchapters.

THE WAYTHROUOH In the Dyad we see the Monad refract as Two. The Dyad emphasizes difference. It foreshadows the world's aPParent boundaries, conflicts and echoesour own senseof separabegins. tion. Oppositesappearwhen separateness Like the line that both connects and separates,the Dyad piscis, unites as weII as divides. Its representative, the "esica envelops a line as well as indicating its center; creating the appearance a division but at the sametime providing us of with the portal through it. The opposite poles of the Dyad retain their memory of the One. At every chance they seek to merge and become whole again. Like a lover, lightning leaps a gap to meet ancdissolve with its opposite charge where they can both retum to a stateof unity and peace. In the metaphor of arithmetic the Dyad reveals itself as the door between the One and the Many, between unity and a1l other numbers. The oesicapiscis was recognized. and applied in ancient art and architecture as a functional s)'mbol of the fusion of oppositesand as a passagewaythrough the world's apparentpolarities. But seeing the Dyad outside ourselves is only half the picture and half the passage.We cannot avoid the polarities within us. The great dual throng of language is evidence of our own investunent in a private netherworld of polarized images. As an exercise, examine the poles within your own thoughts, emotions, and desires. Consider your personal polarities, the situations that strongly attract or repel you, the strong opinions you cherish. Such situations and opinions call us like sirens toward which we unconsciously drift. We rarely resist favoring one pole or another. But orily as we

N u*bu prorreds froffi unity. -Aristotle (381t-322 B.c-)




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Two intersecting circles contain all the proportions necessaryto generate the polygons found in nature artd the five Platonic volumes.

realize that the strength we direct toward one is shared by the other can we jimmy loose from both. By finding their comnon ground, their point of balance, we can pass between them without being pulled toward either side. OnIy then can we experience the journey with the Monad through the z.tesica's dooruray to the further principles under' lying cosmicand human design. Let's continue to use the geometer'stools to assistin the birth of a more recognizable universe. Passing through the Dyad we emerge into the Triad, emissary of the principles of balanceand strucfure.

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Ihree-Prrt l|amol But every tensionof oppositesculminatesin a release, of out wHch comes the "third." In the third, the tension is resolved and the lost unity is restored. --{.arl G. lung (1875-1951, Swiss psychologist psychiatrist) and A whole is that which has a beginning, middle and end_-Arirrorl, A11good things come in threes. -Folk saying The step to Three is the decisive one, which introduces the infinite progression into the number sequence. -Karl Menninger With Three a new element appears in the concept of numbers. I-You; The I is still in a state of iuxtaposition toward the You, but what lies beyond them, the It, is the Third, the Many, the Universe. -Karl Menninger The meeting of two personalities is like the contact of two chemical substances:if there is any reaction, both are transformed. -1arl Jung

Inrnr Th, rriodi, theform of thecompletion aII of things. -Nichomachus of Gerasa (c. 100e.o., GreekneoPythagorean philosopher and mathematician)

Birth of the Triangle. We express the principle of threenessby writing the numeral "three" with a beginning, middle, and end. ?A

Three is the formula of all creatior,. -Honor| de Balzac(1799-1850 Frenchnouelist) , The One engenders the Two, the Two engenders the Three and the Three engenders all things. -Tao TeCh'ing



ONE,TWO, THROUGH Ln the legends of King Arthur and the search for the HoIy Grail---symbol of self-knowledge and self-perfection-Sir Percival was one of orily three knights to actually find the Grail. His name, Percival,derives from "pierce the valley," one who passessafely between looming mountains, Piercing polarifi break:rrrgthroughto transcend opposites, showing the way to inlinite possibilities where there had only been two. Without reaiizing it we "pierce polarity" whenever we count "one, two, tfuee." Counting is a retelling of the original creation myth in the purest archetypal terms. Primitive tribes often count "one, two, many." The Sumerians counted "man, woman, many." But giving separate numbet narnes to quantities past two for the purpose of counting is so for deceptively simple that we take the idea of a sequence granted. Yet our ability to name numbers in an ongoing proIt gressionreflects a major leap of consciousness. gives us the abfity to transcend polar bounds and realize the unlimited. Since "one" and "two" were considered by ancient mathematical philosophers to be the parents of numbers, then their firstbom, "three," the Greek Tiiad, is the first and eldest number. Its geometric expressiory the equilateral triangle, is the initial shape to emerge ihrough the portal of the uesica piscis the first of the Many. , Speak aloud the word "three" in English or most any lalguage past or present and you'll hear its relation to words like "tfuough" and "threshoid" alild the prehx trans (" across,""penetrate"). The leap to "lhrce," as its linguistic root tel1s us, takes us over a threshold and through past polarized limits of the Dyad. Vr'hereverthere are three, as the three knights, three musketeers, three wise men, or three wishes, there is throughness,rebirtlg fransformation, and success. As the first "number" to succeedits "parents," three is in a unique position within the futl sequence ten. lt is the orrly of number of infinitely many to equal the sum of all terms below it (3 = 2 + 1). It is also the only number whose sum with thosebelow equals their product (1 + 2 + 3 = 1 x 2 x 3). Mathematically, these three become one and set the stage for the birth of other numbers.

into AU *n diz.ided three. -Homer (Ninth--eighth century B,c.)




and understanding




geometry/ myth, religion, and natural science will help us recall our inborn but sleeping memory and our deep understanding of the tripartite world and ourselves.

THREE CHEERS! (NOT TWO OR FOUR) We can understand the Triad by noticing where it occurs in our lives. It seems natural to divide any whole into three parts. A beginning, middle, and end, birth, life, and death, the three dimensions of space (length, widt[ height) and time (past, present, future), three daily meals corresponding with the sun's three stages (dawn, noory dusk), phases of a chessgarne (beginning, rniddle, endgame), computer jargon (input, output, throughput), cycle of a traffic light (greengo/ yellow-caution, red-stop), and confrontation in the animal and human realms (" flee, figh| or fortify"). The hiple-goddess religions of ancient agricultural settlements identified the moon's three phases (waxing, full, waning) with the planting cycle (plant, harvest, dorrnant), as well as with the goddess herself (virgirg mother, crone). These ("this, that, and the other") are just some of the ways in which we frame and express what we {eel to be true about the inherent tripartite wholeness of the universe. Subdividing each phase into three parts contihues the process. People structure begirurings with "on the count of three..." "ready, set, go," and ".ready,steady,go," cheer their ongoing with "hip, hip, hooray!" or "hurrah!," which derived from "Hu Ra," the Cossack battle cry signifying "Paradise!" We also end events in threes,asin "going once . . . going twice . . . sold!" and baseball's "three strikes, yer out " a game played in First, Second,and Third World countries. We say something is "as easy as one, two, three" but try saying "easy as one, two" and "easy as one, tr Tolthree, four." Likewise, we 6ay out ABC's, not our AB's or ABCD's. One, two, four, or any other number of steps don't feel right-do they?-when epitomizing a whole process.Fewer than three feel incornplete, and more than three seem superfluous. These values aren't culturally taught but emerge from deep inside us according to our sensitivity to the archetypes

The unity of the circle manifests as a trinjty: center, radius, and circumference.





Icons ald logos. The triangle


is a visually arresting shape and powerful symbol calling for us to be alert. Variations on the triangle have a captivating psychological effect on us that imply a messageof wholeness, strength, stability, and process in commerce, science, and religion.

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within. Three announceswholeness and completion through an embracing synthesis. We feel its corectness and express it through our words and irnages. Three also indicates plurality and supremacy. Drawing three vertical lines I | | and repeating a character thrice were the methods used to indicate "man-y" and "maximum" in Chinese, and ancient Egyptian hieroglyphics ("thrice-great Djeheuti" [ater Hermes tismegistus]). The Egyptians wrote



Alchemical androgl'ne (literally "man-woman" in Greek) symbolizes transcendence by the union of opposites within us.

"thrice dark" to indicate the highesi degree of darkness, "thrice hidden" to indicate the unknowable' We cannot help but use three terms to indicate superlatives like " good,better, best," and "fine, finer, finest," honored by the three gradesof Olympic medals (Bronze,Silver,Gold). The supersiition about not walking under a ladder derivesfrom a caveat{rom ancientEgypt: "Don'tbreak a triangle"; that is, don't disturb a comPleteevent. , Some say that "whatever you do comes back to you three times" and "third tirne lucky." Others say "three on a match" is unlucky. \44ratever the value, good or bad, it's three that we speak of. Religious rituals calling for rePeating a phrase tfuee times, three prayers, or chanting three times, like superstitiously spitting three times, are intended to make an event happen, to make it whole and bring it to manifestation and completion. ' overwhelmingly recurs in reliThreenessas throughness myth, superstition, fairy tales, mysticism, magic, folk gion, iayings, proverbs, and fables, the time-honored expressions that crosscultures. We can discover more of the rnessage of t}reeness by making a list from our own experience. In childhood we leam of Goldilocks and the Three Bears("too hot, too co1d, just right"), threeblind mice,threelittle kittens,and ihree little pigs. In fairy taies and myths we find three wishes, chants, attempts, doors, children, animals, hands' eyes, suitors, wise and weird men and women, hermits, kings, feathers, seeds, sons, brothers, princes, daughters, sisters, princesses, dreams,three'headeddogs, dragons, and other trimorphs, the goddess-trinity, the three Fates (who spin, measure, and cut the thread of each iife), three Graces, Furies,and Charities.We read talesof three gifts, days,jewels,rings, goblets,arrows, swords,keys, gates,Paths,songs, and tasks.Tridents magic words, rivers, judges, challenges, to the greatnessof those they and triple thunderbolts a1lude The s)..rnbol the Greekgod Apollo was the hiof accompany. pod, or three-ieggedstool, upon which the respectedoracle of Delphi sat and prophesied.The GreekgoddessHarmonia (iiteratly "fitting together") was bom of parents Aphrodite (Love) and Ares (War).The triple goddessHecatepresided in over crossroads the form of a cronewith threebodies and



three heads. The letters of her name have values that add up

to 334, a permissible single unit from 333, emphasizing her threefold nature. Consistent themes in these stories emerge quickly: opposites are balanced by a third, mediating element that reconciles a conflict, healing the split of polarity and transforming separate parts into a complete and successful whole. A goal is actualized after two failures. Obstacles are overcome on the third irial. The poles of hero and enemv are transformed by the role of a third person, the helper. Triptychs, or three-part paintings, display the three scenes,or stages, such events. of The high occurrence of temary rhythms, from wishes granted to divine trinities, is more than coincidence: again a number expresses irresistible archetypal principles from deep within us. We respond to symbols of the archetypes outside us to the extent that we're sensitive to their principles buried within our deepernature.


igence *ho* intell

hasattainedto Unity casts away from him bothsin andairtue. -Bhagaaad Gita


THE BIRTH OF THE TRIANGLE If you're ever responsiblefor constructinga universeor part of one, you'll surely need to know how to construct a triangle, the calling card of the Triad. The triangle is the first shape a geometer brings through the opening of the aesica piscis. In constructing a triangle we witness the birth of a shape from an unspeakable archetlpal level into configuration. Every triangle we construct is not merely a copy of another person's triangle but is an approximation of an ideal principle. A11 important points neededto constructa trithe

I heTriad a special hns beauty fairness and beyond all numbers, primarily becauseis theoery it first to make actual the potentialities the of Monad. -Iarnblichus (c. 250{. 330, Greek Neoplatonic philosopher)



Even the impossible reqrrires a trinity. Three planes drawn at right angles to each other are the minimum requited to produce figures like this tribar, the first such design unable to be built in three dimensions.

piscis;1t only remains angle are already aPparent in the esica " for us to connect them. To make the Triad's principles visible, begin, of course, piscis.Connect the cirwith a point. Frolrl that, createa oesica with eachother and with the point above where cles' centers the circles cross.Now welcome the firstbom of the primeval parents, archetypal oPposites, in the form of the triangle (from the Greek trigon, "three-sided"). It is the first area enclosed by straight lines, the birth of surface and structure. The emergent triangle remembers its source in the Monad by having equal sides and equal angles: hence, it's referred to as an "equilateral" or "equiangular" triangle. A triangle is the most astringent shape. In contrast with the circle, which enclosesthe greatest area within the smaliest perimeter, the triangle has the opposite Property: it encloses the srnallest area for the greatest pedmeter. A triangular slice of pizza actually offers fewer toppings than would a round slice with the same pedmeter. Recall ihe experiment in chapter one with a loop of string on the checkered surface. This time verify that a triangle covers fewer squares and a smaller area than any other shape made with the sameloop. The Triad divides the triangle, its emblern, into threes. In all the centuries, geometers have still discovered orrly three types of kiangle: equilateral, in which all three sides and angles are equal; isosceles,in which two sides are equal but the third is differen! and scalene,in which no sides or angles are equal. Likewise, al1 the world's geometry finds only three types of angle: acute, or less than ninety degrees; a right angle equal to ninety degrees; and obtuse, or greater than ninety degrees. A triangle is a statement about relationship and balance. As the centers of the two circles repel and tug at each other, a reconciling third point occurs naturally above the place where the circles cross and agree. Thus the ancient mathematical philosophers referred to the Triad as prudence, wisdom, piety, friendship, peace, harrnony, unanimity, and marriage. We see ttris balanced path between extremes in periodic phenomena such as a swinging pendulum and a heartbeat. The birth is not yet complete. We can bring more of the piscisby extending the lines from triangle through +heaesica



the crossing poin ! down throughlhe centers until they reach the opposite sides of the circles. Connect those points with a horizontal line to complete the large triangle. Notice how this line passesprecisely through the point where the circles crossbelow. Connectthat point to eachcircle's centerto fill the large triangle with four equilateral triangles. Ancient philosophers were struck by the ptofound harmony of the oesica's space, which generates triangles by its natural Doints. The mathematical relationships between the sides of a piscis, which. gives birth to it, can be triangle and tLreaesica made more comprehensible by making them audible. Sirnp1y do the conskuction on a wooden board. Then hammer nails halfway in at the points that make the triangle and at the circles' centers,Tie a single guitar string or piano wire to one nail, and wrap the remainder around each of the nails. Tie the end tightly to the last nail. Pluck the string between eachpair of nails to hear a tone. Equal lengths should produce equal tones. Pluck the length of the triangle's height and the distance between the circles' centers in sequenceor simultaneously to hear the significance of the mathematician's pfuase "the square root of three to one," the seedratio of the equilateral triangle. THE ARCH OFEXPERIENCE Designis but anothernamefor natural law. -Moses benMaimon(Maimonides) (1135-1204, philosopher) Jewish VVhendesigning a universe you'll need triangles to create self-supporting structures. The triangle is, in fact, the definition of structure since it is the only polygon structurally rigid by virtue of its geometry alone. It is s1'nergisticin that its stability and superior stength are unpredicted by any of its parts, which, by themselves, do not have theseproperties .

tf Surprc iscornposed triangles. -Plato



How an architectural A-frame truss supports weight.

How an arch diverts weisht.

Our pelvis is the keystone supporting our body.

An orrhnezter sleeps. -Ancient Hindu aphorism

A close look at the fundamental structures of the world shows they're bracedby triangles. Triple-structured architecture, including arches, is fundamental to natural and human designs. Urilike any other shape, the three sides of a triangle resolve opposite tensions into one solid, stable whole needing no support from without. A triangle is self-sufficient. ln an arch, the third element, the point atop the triangle where the circles cross,is the keystone that diverts weight downward around its opposite sides into the earth. In a sirnilar way, our pelvis is the keystone atop the triangle formed by our legs. To feel its strength, stand with your legs slightly apart and notice your torso balancedat the pelvic keystone.Architects and sftuctural engineers employ the principle in the omnipresent A-frame truss. Carpentersknow to stabilize a structure by bracing it with triangles. We see the same principle in the curve of insect shells, dams, tunnels, bridges, and monuments like the Eiffel Tower. Triangles bestow strength, balance, and efficiency of space, energy, and materials. Fold a paper into an accordion shape and seehow triangles support the weight of an object. The more triangles, the more weight it will support. Triangular structure gives the rose's thorn and shark's tooth their bite, and the wedge and ax their splitting power. Giassblowers have long known that making an arched



indentation at the base of a wine bottle strengthens the whole container wtile also collecting sediment. We incorporate triangles into our creations for reasons of superior structure, strength, efficiency, balance, visual appeal, and s;rnbolism. Discover the Triportite Pottern in Noturol Forms Life-forms in general have a three-part structure. From the body parts of an insect (so named becauseit's "in sections") to the hunian body's head-torso-legs with their own subsequent tripartite subdivisions and the three layers of heart muscle, all express the same principle. The geometry of fruits and vegetableswill be tripartite when they begin as three-petaled florniers.Observe before you eat them. If you're going to conshuct a universe you should also know how to dissolve it. This is done by learning how to create cracks properly. When we look at cracks in a sidewalk we're actually seeing microscopic lines of force magnified rnillions of times. hr nature's view, concrete and rocks, drying mud and coffee grounds left in the filter are elastic materials. They crack relatively "suddenly" over moments, hours, days, and months along three-way 120-degreejoints because this is the way their coequal molecular bonds tear apart rapidly like an A-frame in reverse. If you have more time-years, decades,or centuries-you'll seematerials like old paint and porcelain crack more slowly, resulting in an alternative, four-cornered pattern (see chapter four). Prac-

[An archis]two which together weaknesses make strcngth. a -Leonardo da Vinci

L OrCe Wttnout wtsaom


falls of its ownweight. -Horace (65 B.c.-88.c., Roman poet and satirist)

Sidewalk cracks display three-comer joints.



tice reading the different crack pattems by noticing thern in familiar places. A triangle can be geometrically "decomposed" or subdivided into smaller triangles that diminish by the same root-3 relationship. Follow the instructions below and decompose a triangle. Then superimpose this geomeky onto the illustrations of the flowers, fruit, microlife, and insects on the next page to reveal their intemal tripartite design structure Within each z.tesica piscis, a large triangle has been constructed.Decomposeit to seeeachlife-form's design.Some have been done to show you how. Every triangle can be subdivided into four smaller triangles. For example, further subdividing the upper triangle of the fly diagram will show you how its head is formed. Do this construction on the images of other tripartite natural objects you encounter. Many fruits and vegetables display three-comer structure.

Thrrri, onunfoldingof @A theOneto a condition where canbeknownit unity becomes recognizable. -{arl Jung



Construct a ?resica pisclsand a large kiangle within it. Subdivide the-tdangle into smaller triangles. Draw three lines connecting each of the triangle's comers with the midpoint of their oppoJite sides.The points already exist.The lines meet at 120.. Those three lines cross the innermost triangle at points that can be connected to make a smaller but inverted triangle. Repeat the process cormecting each new set of three points where thl lines crosseachsmaller hiangle to draw the next hiangle. Contlnue constructing smaller triangles as your sharpened pencil allows. A1so,extend the sides of the inner triansies to seehow triangular structureperpetuatesitself inward lyi



Honeydew melon

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THE ONE THREEREMEMBERS The geometry of knots, with their over-under crossing pattems, is of great interest to modern mathematicians. The simplest possible continuous-line knot, the trefoil knot, itself in three places.No knot can have fewer than crosses three crossings,the minimum required to complete a whole. The word trinity, rn fact, derives from "Iri-urrlty," or "three as one." The mystery of three-becomes-one can be studied through the sign known as Borromean Rings, named for its appearance on the crest of the Borromeo family of the ItaIian Renaissance.It provides an interesting metaphor illustrating the essenceof the Triad. At first glance the sign seems to consist of three intertwo locked rings. Yet on closeinspectionwe seethat t?o rings actually interlock:the white goes oaer the gray, the gray goes ouerthe black, and the black goesooerthe white. Alone or in pairs they're separate. Each rnaintains its distinct identity. But when all three come together they lock as one, Iike a trefoil knot, a self-supporting whole beyond what the parts could do alone or in pairs. It takes all three interlocking simultaneously to become unified. Cut any one ring and they'll all come apart. Construct a set of BorrorneanRings. You'll need paper, cardboardor index cards,compass,and pencil. The Triad's retrograde connection to unity recurs in the braiding of hair. Ordinary experience shows that two tresses twirled around each other won't hold together but will soon unwind unless they're tied at the bottom, That is, the Dyad's principles wiII not create an enduring whole but merely opposites without resolution, an unending dance of conflict, Iike lawyers without a judge. But introduce a third strand of hair and altemate the three strands to weave a braid, an ongoing trefoil knot, a self-reinforcing stable unity. A braid's three separate tresseslock as a single new whole like Borromean Rings, needing no outside tie, self-sufficient and stable from within. A bow is only for decoration. This is a clue to the archetype of the Triad: a properly chosen third factor induces a relationship between opposites that unifies them and brings them to a new level. Threeness resolves the Dyad into wholeness and unity, back

Trefoil knot

A rnor|ota is not cord quicklybroken. -Bible (Ecclesiastes 4:12)

Braided hair weaves a threein-one whole.

51 (1) Make a point and tum a circle.(2) Then slightly closethe compassand, using the samecenter,tum a smallercircle.(3) Repeat this three tirnes to make three rings. Color each ring dif{erently. Cut them out, and then also cut their irner circles. Snip two rings. (4) Study the illustration and carefully slip the rings together to unite them. Tape the snipped rings closed. Verify that although no two dngs are actually linked, all three together form ar-r inseparableknot. Three become one, exfdbiting the pdnciple of triangulated structure.The step to threereestablishes ,,lost,, the unity deniedby ihe Dyad, resolving Twoness back to Oneness.



\7 /A


oo@) Lighting six, or twice three, braids of the Havdalah (Hebrew for "Differentiation") candleat the closeof the Jewish Sabbath represents the act of discriminating twilight from day and night, and human life woven between the sacredSabbathand secular weekdays.

through to their source in the al1-pervadingMonad. The embroideredcosmosis braided into existence. the Native In American languagesof the Nez Perc6and Shahaptian,the number threeis known asthe "centerof the one.,, The Triad pervadesour lives. The principle of the Triad as seenin the braid, that a minimurn of threeelementsis required to weave into a who1e,teaches how to reconcile us con{lict. No enduring resolution of any kind is possible without threeaspects, opposites two and a neutrai,binding, balancing, arbitrating, transforming presence.Knowirig how to choosethe third factor meansihe differencebetweei a conflict's resolution and its perpetuation. The third element, if properly chosen,"pierces the valley,, and achieves a previouslyunknown level of experience, one of balance and completion. example,if I cin,t decidewhetherto go For out or remair indoors I might consider another factor to induce a decision:the weather,my friends, responsibilities, or needs. Similarly,we act as relativeoppositesto some people and havean arbitratinginfluence bltween others. our per_ If sonal and professional relationships could be visualized graphically, they might appearasa Fersidn carpetwoven of braids within braids. We can also use an understanding of the Triad to avoid the lure of the dual tJrrongof polarizid language.Sincethe



The principle of the Triad is sumrned up in a simple hanging balance: two opposites are reconciled by the presenceof an independent third aspect above them both. This is why scaless).rynbolize justice by frlal.

words and grarunar with which we think help to shape what we perceive, seeing the world in terms of black-andwhite opposites and either-or situations detours us to one side of a valley or another. But the Triad invites unlimited perspectives and possibilities, infinite shades of gray between the extremesol "yes" and "no," eachvalid in different situations. We're constantly exposed to threeness. Every whole event is inherently comprised of a trinity of two opposites and an outside third element that brings about a new whole. Physicists cail this trinity an "action, reaction, and resultant"; philosophers call it a "thesis, antithesis, and synthesis." The three elements together form a greater new thesis, which, in tum, induces its opposite and is ready for a greater synthesis. In becoming whole the Tiiad, threeness, rernembers unity and expressesits yearning to return to the state of the Monad. The following examples eachcomprise a triune whole, a Borromeanknot or triple braid. Use them to leam to recognize the three components in any whole event. Begin by dis-

OpposrrE Poues Seller Man Water Proton Prosecution Cook Artist Projector Management Wick "North" Pole Negative Reactant Left Pan Buyer Woman Flour Electron Defense Eaters Medium Screen Labor Wax "South" Pole Positive Reagent Right Pan

BrNorNo EueveNr Money Love Fire Neutron Judge Food Inspiration Film Arbitrator Flame Field Wire Catalyst Central Hinge

NewWuoue Commerce Home Bread Atom Trial MeaI Art Movie Mediation Candle Magnet Electric Circuit Chemical Process BalanceScales





This utorldis a aast unbroken totality, joins its A deep soli.darity contrarypowefs. -Sri Aurobindo Ghose

Helium, the second atom to manifest,the fust elementcomoosed of threedifferent "charges." tinguishing the two opposites, and then note the cataiyst or binding element that causesthe new relationship to arise. Tripartite architecture braids the cosmic tapestry. Albert Einstein's famous equation E=mc2 describes the whole cosmos as Borrornean Rings of three distinct but r.rnified aspects of energy, mass, and light. Modern mathematical physicists describe mass,commonly called matter, as atomic whirlpools. After the emergencethrough the polar portal of manifestation, all subsequent atoms beyond hydrogen are composed of three "parts," like spiruring Borromean Rings of positive protons and negative electrons balanced by neutral neutrons. Atoms configure in pattems along lines of force whose surface features we smooth out and simpiy call "things." Color is also based on threes. Direct liqht emanatesfrom an iJluminating sourcelike the sun, a stai, or television set. Pure whole white sunJight is itself a threefold weave of primary colors (red, greerl blue-violet).Light's primary coiors are "additive" becausetheir combinations braid together to lecgmpgse whole white light. Verify this by shiniirg ttuee flashlights, each with a different primary colored fi1ter, at one spot on a wall to produce white. Look closely at a color TV screenwith a magnifying glass and you'll seethe three primary color cellswhoselight blends in our eye to form the colors on the fube's suface. Pigment colors such as those in paint, ink, pencif crayons, and chalk occur by "reflected light." Their three pri-


Pigment Colors of light and colors of pigment build in opposite directions.



Shinto symbol of the tripartite revolving universe.

mades (red, yellow, blue) are called "subtractive" colors becausethey iemain afterd\recllight colors are absorbedinto or "subtracied" from the surface. Just as direct light adds toward white, Pigment colors subtract toward black, the absenceof color. The reflected colors of the obtectsaround us show us exactly what the actual colrorisn't becauseits true colors were absorbed. $Ihat reflects to us is what remains' The ink on this page is black becausea1Ilight is absorbedinto it or subtracted and what remains is no color at all. The principle of the triple weave governing colorfuJ light vibrations occurs also in sound and music. It takes combinations of on-ly three primary notes (the tone, fourth, and fifth) to cornbine in four secondary ways to produce the natural seven-notemusical scaleof a vibrating string or column of air (seechapter seven).The triple weave is the basic knot of this embroidered universe at each level of matter, energy, and light. The fact that human body structure is three-part shoulcbe no surprise since the body begins life as a microscopic three-part stmcture. At three weeks, the human embryo has (Creek enfolded spherecalled a blastula becomea sixty-ce1l distinct layfor "bud") that enJoldsand diversifies into three ers to become the different but intertwined systems of our body. We may think of our skeleton as our body's solid unchanging core of bone. It shouldn't change muclg should it? But like other living cells it embodies a d1'namic tlreephase cycle of transformation: emergence, development,

ffi ffiffiffi@ Embryological development of a vertebrate anirnal.





+r1 0stEocytss /E\



988 The thee t,?es of bone ce11 correspond with the three Hindu deities, Brahma, Vishnu, and Shiva/ perpetually creating, maintaining, and destuoying.

.=K |.ar. and dissolutiory or, simply, birth, growth, and decay. Thus there are three types of bone cell. Tlre osteoblast (Greek for "bone bud") createsnew bone and repairs old ones bv extracting calcium from the blood and laying down a bone mattrx. Osteocytes("bone cells") maintain and repair the bone's substance and strength.An d osteoclasts ("bone breakers") dissolve and reassimilateold bone materials by dissolving bone matrix into the blood. Osteoblasts reabsorbit to make new cells, and the cycle continues. The ancient trinity of gods of the Hindu religion called 'liimurti (Sanskrit for "one whole having three forms,,) mythicaliy personifies the functions of the threefold bone cycle: Brahma the Creator, Vishnu the preserver, and Shiva the Destroyer. Myth sometimes cannot help but be sciencein orsgu$e. The sameprinciple of tripartite unity was consciously intended in the creation of the government of the United States. \4/henits founders were d-eciding how to construct a stableyet dy.namic and enduring systenithey looked to their background as Freemasons.Esoteric Freemasonry of the eighteenthcentury usedarchitecturalsymbolism to convev ancient teachings of self-knowledge ana inner development passed down from Egyptian times through a chain that included "Peter Gower" (a thinly disguised name for

/H, tI



T I assure I hadrather you excelothersin the knoraledge what is of excellent than in theextent of my poweranddomain. -Alexander the Great (356-323 B.c.)



art nature, my T hou, to goddess; thy lawsmY are seraices bound. -William ShakesPeare

Pvthagoras) and the Gothic cathedral builders' In designing the go:vemment the founders devised a democratic system but of cliecks and balancesamong three seParate interlocked is branch, or presidency, the Borromeanbranches: executive intended to clash with its etemal oPposite, the legislative branch of Congress. Their conflict is supposed to- be balanced by an ideally neutral judicial branch,- the Supleme Court, like the neutron. of the atom, which has no biased charge but allows the protons and eleckons to interact creatively as a whole atom. The three-part federal-braid recurs in staie and local governments, each a smaller braid woven within the tripartite whole. Nature tdaches,technology builds, and we know deep within us that three can become one. We accept this concept without thinking when we sit on a three-legged stool' \iy'hat two legs cannot have done, a third perrnits. Three brings a p(ocess to completion' But this achievement is not necessari1y static. It can set into motion a triangular cyc1e,an ongoing rhythm of birth, growth, and dissolutiory a tfuee-phase priceis with many fices in which all change must partake' i,Vhether an arch transmits weight into the earth or bone cells change along a cycle-and even in the greatest cosmic transformltions of light, energy, and mass-the weave is threefold.

Praver of 120" forms a trinity.

TRIUNESELF The triangle is the world's Preeminent symbol of divinity. As the symbol of holy trinities, it reachesinto nearly every religioui tradition. In prayer our hands spontaneously form comer.The urge of the Triad is very a three-sided120-degree powerful because it is rooted in our own triune nature, which is modeled on the world's. Long before Einstein phrased E=mc2,ancient geometers represented the trinity of light, energy, and mass by theii tluee tools: the compass with its unsleeping eye or sun above and its legs asrays of wisdom and beauty shining into our lives, the straightedge that directs energy Patterns of tension and movement, and the pencil and paper that make the pattems visible. Thesethree tools of light, energy, and massby which the



Divine geometer constructs the cosmos and by which the symbolic geometer approximates archetypal patterns are also mirrored in us. \A/hat scientists call ,,light, energy, and mass" are the traditional "spirit, soul, and body" described by Plutarch as rors (divine intellect) , psyche(soul), and soma (body). To the Hindus the principles within light, energy, and mass are the three gunas ("qualities,,) of purity, activity, and inertia that blend in different proportions in ail processes and eventsoutside and within us. Our "spirit" may be thought of as direct light, the wisdo_m a1$ beauty of our deeper Self, whose knowing is reflected on the waters of our psyche as our ability to think. V\{hatis deeper within us as delight is reflected as emotion. \4hat is deepest within us as the most mysterious ,,Will to Be" manifestsas our most explicit structure, the body with which we act. Freemasonscontinued this geornetric architectural symbolism in the new govemment. The United Statesis one of a few countries whose official seal has two sides, both on the dollar bill. The pyramid on the reverse,with its luminous triangr.rlar and trapezoidalbasebelow,is an abstractmodel eye of ourselves. The eye represents our ever-awake, deep, divine spiritual Self, the thick foundation below its outwaid expressions,our energetic psyche and denser body. Note the kapezoidal shapeof the base.Jt,ssimilar to a design_often seen as an Egyptian sphinx,s or pharaoh,s nemesheaddress, whose slanted sides, like the bise of the pyrarnid, imply an unseen friangle above it representing our little-suspecteddivine nature (next page).


Saintshavehalosof different shapes, orrly God'shalo but is traditionally a triangle.

@ @


DISCOVER THETRIANGULAR PATTERN IN EGYPTIAN JEWLRY The ancient Eglrpfian priest-craftsmen used geometrv extensivelyin their designs,both struchrally and-sr.mboli_ cally. Simultaneousrepresentations Self and the world,s of strucfure were often incorporated into their art, crafts, and Eighteenth-century French architecture to remind the viewer initiated in its system of omament. s)'rnbolsof their significance.



Pharaoh AmenhotPe

Egyptian blue faience bowl. A point in a circle, the hieroglyph for ihe sun and light, symbolizes the supreme spiritual principle. It sits at the center of a divine triangle as the common eye of three fish swimrning in the waters of our psyche. As in Hinduism and Buddhism, the lotus implies unfolding awareness. This bowl was an everyday reminder of ihe divine triune light at our center.

The trapezoidal shaPeappears often in Egyptian jewelry. The pectoral of SenwosretII (c. 1890B.c') (next Page) was designedas a trapezoid to fit perfectly at the bottom of piscis.In an equilateral triangle constructedwithin a uesica typically subtle Egyptian slnnbolisrn, what is not visible and by beyond name or form is best represented empty or "neg"top" abovethe hapeative" space,in this casethe missing zoid. Unseen divinity is implied by the "empty" ftiangular spaceabove the visible form. upon the image A geometricconstructionsuperimPosed will help us seehow it was designed.Notice how the elenents of the composition Iollow the prominent points, lines, and areasof the construction. The center of the whole triangle is precisely at the tail of the scarab,where the sun or spiritual principle is bom. To an initiate who can read its symbols, this pectoralis both a cosmologicaldiagram and a map of our Se1f.Gllphs of life, time, and eternity are encompassed within the divine triangle. This jewelry, Iike much of Egyptian ar! honors the divine spacewithin each of us. Further explore the rich geomehy of this pectoral by extending lines and connecting obvious Points.


The principles of the Triad are omnipresent.Takingus past the oscillations of polarity by synthesizing a new



whole, the function of balance appears, natural forms -sequentiallv become stable structures, and ordered processes actualize.We'll seethis trinity of ,,shucture-funcfion-order" and "space-power-time"aiso expanded in the principles of numbers that are multi plesof three,in chapters six and nine. Religiory myth, folk tales, science and art tenaciouslv reiterate the importance of trinities due to their deep roois within every part of us. They have powerful psychoiogicat ettectson us because tripartite universeconnects the with its archetJ4)alroot within us. Threenessshows us that we,re not separate from the rest of the universe but are literallv braided into i! we are a complete whole living in a greater completewhole. Yog already knew all this but may have had to be reminded. Now halring the ability to express the geomehic con_ structionsof the Monad, Dyad, and Triad, we can represent the point, the line, and the surface.The principles of num_ bers next lead us onward to give our universe depth. Going, gong, gone.

Pur, ,rrrnrr, andpure mntter, thetwo and joined one into were shot forth without flaw, Iikethree brightarrows from a three-string bow. -Dante


tu un

J'/|ilherStlh One carmot help but be in awe when one contemplates the rnysteries of eternity, of life, of the marvelous structure of reality. -Albert Einstetn . . . to know the order of nature, and regard the universe as orderly is the highest function of the mind. -Baruch Spinoza (1632-1677Dutchphilosopher) , The perfection of mathematical beauty is such . . . that whatsoever is most beautiful and regular is also found to be most useful and excellent.


-Sir D'Arcy Wentworth Thompson In all their works [the ancients] proceeded on definite principles of fitrLessand in ways derived frorn the Truth of Nature. Thus they reached perfectiory approving only those things which, if challenged, can be explained on the grounds of tmth. -Vitruaius (Firstcentury8.c.,Roman architect, engineer, witer) and What canst thou seeelsewhere which thou canst not seehere? Behold the heaven and the earth and all the elements; for of these are all things created. --Thomas h Kempis(1380-1471,Germanecclesiastic writer) and We carry within us the wonders we seek without us. -Sir ThomasBrowne One race there is of.mery one of gods, but from one mother we both draw our breath. .-Pindar (c.522438 a,c.,Greek lyric poet) The Birth of the Square. 60



OFTHE DEPTH IN THE SEARCTI Although the ten mathematical archetypes exist simultaneously, it can help to view them as a sequence.We've seenthe creating process emerge from the Monad's point to the Dyad's iine and Tiiad's surface, that is, from zero dimensionsto one dimension to two dimensions.Our joumey corresponds with the emergent structure of a planl the point of the seed,the line of the stem, the surface of the leaf. We now come to three dimensions, to volume, to the fruit of our geometric olant. Three points define a flat surface, but it takes a fourth to define depth. The archetype of foumess, called Tetrad by the Greek mathematical philosophers, is expressedin geometry and nature as oolume,three-dimensional space. The Tetrad provides the framework of anything we can point to. Like other mathematical archetypes, the principles of the Tetrad are bom as shapesthrough the portal of t}:.eoesica piscis. But now triplets emerge: the tetrahedron, the square and the cube. The flat square or tetragon (Greek for "four (Greek for "four sides") has four equal sides. The fetrahedron {aces") is a volume of four identical triangular faces.And the ("six faces"), the most well known volcube or hexahedron ume, is enclosedby six squarefaces.These forms recur in natural and human creations and symbols. The geometry of the Tetrad mani{ests itself in nature with greatest exactnessat the borderline between nonliving and living forms. The three geometric forms are widespread as the remarkable subrnicroscopic structures of atoms, molecules, crystals, viruses, and living cells. Let's seehow the Triad's flat surfaceblossomsupward to become the Tetrad's volume, how the triangle becomes the tetrahedrory the fust fruit of the creating process.

<v Ll tb



THE BIRTfi OF DEPTH Construct the Tetrohedron Here'show to bring depth to the universe. (1) First constructa oesica piscisand large triangle. Subdivide the triangle into four srnallertrianglesby connecting (2) the circles'centers and their intersections. Draw tabs

Hast thouerfiered into thesprings thesea? of or hastthouwalked the in search thedepthT of -Bibie (Psalms38:16)



so'Iids TransparentrZ\



Four spheres stack as a tetrahedron.

Insects walk on three legs at tune.

extending along the three outer triangles as shown for gluing. Cut out the form around the tabs. (3) Creaseeach line and fold the three corners upward until thev meet at one point. Glue the tabsalongthe insideof eachedge. Hold the tetrahedron and rotate it. It appears similar from every direction. The tetrahedron is the only threedimensional shape whose comers are the same distance from eachother.Other than the sphere,the universe doesn,t allow any volumeto havefewer thanfour corners, faces. or Pressit between your two hands, between a corner and its opposite face,to feel its strength. Of all structures,this first three-dimensionalshape is the most spare. Spaceis desiccated to its minimum volume. Buckminster Fuller considered the tetrahedronto be the basicunit of space.\44ri1e the sphere enciosesthe most volume in the leait surfacearea, the tetrahedronencloses leastvoiume with the greatest the reiative surface area. Being the minimal solid, ii is the skongest and most stable. The birth of the tetrahedron, the first volume, brings the tiad's principle of triangular stability to three dimeniions using the fewest materials. We find the Triad in tripods, musicstands, three-legged stools, campfires, and wheelbarrows. The tricycle's three wheels form a triangle on the ground to support the seaiabovewith greateststi'bility. The occasionaltetrahedral packaging of milk and juice takes advantageof the volume's great strength. Insects make use of tetrahedral stability for walking. Having six 1egs,they walk on three at a time, alternatin-s every other leg from one side to the other. They move forl ward as if on tripods.The lotus positionof yoga stabilizes the body into a self-supporting pedestal for-the mind,s undistractedmeditation.



There are always four ways to look at any three-dimensional structure: ai points, lines, areas, and volume, or as comers, edges,faces,and from the center outward' This construction bJ t a tetrahedron of four faces.We can also think of it as four corners,made by simply stacking four spheres like oranges whose centers make the corners of a tetrahedron. In the 1870s the Belgian physicist Joseph Antoine Plateauconstructeda tetrahedral wire frame and dipped it in soap solution. To his amazement the film did not iust cover the facesbut met inwardly to createa fascinating internal structure.After measuringit he discoveredthat the film followed the path that made it cover the minimal surface area possible yet connect every edge. $/hen he then caPtured a single bubble in it he was further astonishedto seea curved tetrahedron at its center. Wire frarnes are a sma11, way for nature to reveal her own rninimalist geometry, the one that shows up as microscopic forms, the strucfures ol molecules, crystals, and microlife. Architects have found that by building along the lines of force shown by Plateau's soap films they can create extremely thin yet strong stuuctures that resistscompressionand tension. Bend wire to form a tetrahedron and extend one edge to use as a handle. RepeatPlateau'sexperiments to seehow forrn in nature is a graph of forces.

in *oy A ,hing ,ndure if nature it is duIY to proportioned its necessity. -Fra Luca Pacioli (LM5?-1514? Italian , mathematiciary writer, and math teacherof Leonardoda Vinci)

as N oturc uses littleas of possible anythinS. -Johannes Kepler (1571-1630, German astronomer)

A wire tetrahedral frame dipped in soapsolution reveals the geometry of minimal forces, minimal volume, and maximum strength, expressedin nature as the silicon skeleton of this microscopicNesselarian.



o The tetrahedron commonly occurs in organic and inorganic chernistry and in the world's subrnicroscopic structures. Its geometry frames the architecfure of many elementary molecules, including rnethane (CHa), ethane (C2H6)and ammonium (NH3), the basis of amino acids, the building blocks o{ life. In each of these three molecules, a carbon or nitrogen atom sits at the center of a tetrahedron at whose four corners are smaller hydrogen atoms. The similar charges of hydrogen atoms' electron pairs causethem to go asfar as they can from eachother. The result is a tetrahedron, the only three-dimensional shape whose every comer is the samedistancefrom every othei comer. Substancesmay be composed of identical atoms whose different geometric configurations give them different characteristics and properties. Glass and quartz, for example, are both composedof sand, silicon dioxide (SiO2). Glassis brittle and transmits visibie light while blocking ultraviolet rays. The unstructured molecules of glass flow as a liquid; in fact, glass, the oldest synthetic polyrner, is not considered a solid but a supercooledliquid flowing slowly over decades. (Notice how the bottom of an old window settles in thick, wavy lines.) But the same SiO2moleculesin quartz link so that every silicon atom is at the center of a tetrahedron with oxygen atoms at its four comers. Its extended tripod structure makesquartz a hard crystal,which,like glass,transmits light, including ultraviolet frequencies.

6 Methane Ethane


Carbon tetrachloride



Two views of a diamond showing its carbon atoms linked as a tetrahedral skuctwe. The short bonds between atoms insure the diamond's great strength.

But quartz is not as hard as diamond, the world's hardest naturally occurring substanceand best conductor of heat. The diamond is also structured as a tetrahedral network, but only of carbon atoms, eachof which sits at the center of a tetrahedron having four other carbon atoms at its corners. Becausethe energy bonds between the atoms are so short and strong, lying along the edges of a minimal tetrahedron, a diamond is firm and difficult to cut. But diamond cutters know how to overcome its hardness by slicing along the tetrahedral faces. With the Tetrad come volume and structure. But why is the number four associatedwith physical manifestation? To better understand the Tehad let's examineits exptessionas the square.

I{ umb s,fur ther e, er mor asarchetypal structural constantsof the collectizte unconscious, possess a dynamic, actiae aspect which is especinlly importantto keep mind. in It is not what we cando with numbers what but theydo to our consciousness is that essential. -Marie-Louise von Franz flungian psychologist and writer)

BACK TO SQUAREONE The squareis an obvious symbo] of fourness.Unconscious knowledge of it is so strong within us that it wells up in many of our sayings. We speak of getting a "square deal" and a "square rneal," of "square living," being "fair and square," "facing problems squarely," and sometimes going "back to squareone" in order to "squareour accounts."If an event doesnot "square with our senseofjustice," or a story is "not squaring with the facts" we sometimes must have "square-jawed determination" and "stand foursquare," having our feet planted "squarely on the ground." Like its sibling, the tetrahedron, the squareis associatedwith equality, reliability, faimess, firmness, and solidity. "Four, firm and fair" goes the saying.



I haae keptmy square, not but that to come ShnIIbedoneby therule. -William Shakespeare

We write the numeral four as if in 2x2 gritl.

ffiffi L r, nora betruly h excellent, -square in four handandfoot andmind,

without blemish. formed -Simonides (55G468 a.c., Greek poet)

The ancient Greeks noticed that four is the first nurnber formed by the addition and multiplication of equa\s (4 = 2 + 2 = 2 x 2) and so was consideredthe first even number and first "female" number. (Even nurnbers were considered female, odd numbers male. In Hebrew and other languages, the namesof paris of ihe body that occur in pairs, like eyes and hands, have a feminine ending. Words for parts of the body that occur singly, like the nose, have masculine gender.) To the Pythagoreans the even-sided square represented "justice" becausefour is the first product of equal values,a number divisible every way into equal parts. That is, 4 = 2 + 2 and the two twos can be further divided equally (1+1) + (1+1)and eachdivided again finally to 1+1+1+1. Four is the first number to reach back to its source in the Monad this way. The planet earth below us, solid ground, terra firma, is the supreme s1'rnbol for substance,mass, volume, strength, and stability. We organize space on the ground by the four cardinal directions of the compass and our body, dividing the circle of the horizon around us into four qualters in front and backof us, to the left and right. The four-comeredcross in a circle has long been the astronomical symbol for planet Earth. The four kaditional winds blow across the four corners of the globe. We quarter not only spacebut time/ naturally dividing the year into four seasonsbased on the relationship of the sun with ihe earth around the two equinoxes and two solstices. Foumessindicates associationwith the stable,solid earth. The ancient Egyptians syrnbolized this by showing four pil-



lars arising from earth to support heaven. The Mayans likewise portriyed four beings as supporting the celestialroof' Regardlessof the culture, the square was the Preeminent symbol for the ancient Earth Mother goddess.The association of the earth with the number four, femaleness,and justice is very ancien! far preceding recorded history. The prinshe ciplesof the Tetraddescribeher nourishing aspect: gives The ]apanese glyph for her creations with material substance, and birth, clothes "earth" encourages their growth equally' The word nature comes "country" is a square the enclosing {rom the Latin for "birth." ln the Navajo language, nature rs syrnbol ideograms for "people" and called "Changing Woman." The Navajo term for suruner/ "protection." then, is "Changing Woman's happiness." Mater, the La$n word for "mother," has given rise to the word "matter," also related to meter(" measure")and matrix.The Latin word for father, pater,gave rise to "patrol" and "pattem." Thus, natural forms can be seen as coming from both a mother and father, the mating of matter and pattem. In the way that a mother provides nourishment for her developing child, the "world mother" is described in myths asnourishing the emergent archetyPal pattems with matter, consisting of four tlpes of food, called the four ancient e1ements: earth, water, air, and fire. TheseelementsprovidecAncient Vietnamese "four the mythopoeic way of referring to the modern scientific perfections" coin symbolizes four statesof matter: solids, liquids, gases,and plasma or the square "earth" within the electronic incandescence.The three denser states are familiar to everyone as ice, water, and steam. The fourth state, circle of "heaven." which the ancients called "fire," is known today as "plasma," the glowing electrified gases that "bum" in the sun and stars and cause the fiery glow within fluorescent Iights and neon signs. We laugh at the simPlicity of the ancient concept of just four "elements" comprising the world when today we recognizeover one hundred varieties of atom. But no matter who looks at the world or when, we of can find only fow phases mater,four clothings of nature. Nature unfolds thesefour statesin a particular order.Ice must pass through its phase as water before it becomes steam.If you ever forget the correctorder Plato gives in his just hll ajar with water, and soil, and air, and shake Timneus, it up. The natural layering into which it settleswiii show you This square chessboardreprethe order of the elements, or states:rnost dense earth on the sents material interplay in the bottom, then water, above that air, and, most ephemeral, world. light passing through.



Labyrinth at Chartres Cathedral, and Italian Renaissancegarden p1an. Square labyrinths and thosebuilt around squares and crossesare metaphors of our wanderings through the earth, syrnbolically within ourself, and our path of hanscendence,

Being so close to the earth, we express the archetypal s)'rnbolism of the Tetrad in many ways without realizing it. In cities we walk on sidewalks inscribed with souaresthat symbolize the earth they cover.As communities grow they are often laid out by city planners on a "rational" squaregrid street pattem. We build right-angled architecture and erect chain-link fences, made of tilted squares, as s1'rnbolic The chessboard the is walls of earth. Beds,tables, chairs, and thrones are miniature uorld, the pieces the are models of the earth.The squareinsistson serwingas a matephenomena theuniaerse, rial boundary. of Gridded graph paper is a symbol of the "four corners" and therulesof thegame of the earth, the largest drawing surface there is. Farmers arewhat we call thelawsof piow square fields on the traditional forty acresof land. The Nature. ancients noticed that four-footed animals are earthbound. Four shows up whenever there is reference to earth and -Thomas Henry Huxley matter. A baseball"diamond" is actually a square,asis a boxing "ring" where fighters "square ofi." Many sports fields are square-cornered, signifying arenas of earthly action, material conquest,and physical interplay. The history of board gamesis full of square-comered fields of play symbolizing initial equality in the earthly sphere of action, mass, and force. In classicaiorigami, the Japanese of paper folding, art each construction always begins with a square. Thus, origami cranes,horses,frogs, plants, and people all sharea common mateL The harlequin's costume with its checkerboard pattem s)'rnbolize his role as a chthonic force. There are four traditional colors in the tarot deck (white, black, red, green). Ali quaternaries indicate the material and perceptible. Hopi qrmbols for Mother Words containing the letters "qu" often hint about their Earth.




relation to the word "square" and its Latin counterPart "quadratus" as in quartet quartet, quart (one fourth of a gallon), quarry (where square-comered blocks are cut), quarantine (forty traditional days of isolation), and squadron (a group of four). In the classic s)'rnbolism of m)th and religion, the number forty (= 4 x 10)marks a passingbeyond (seechapterten) a worldly or fourfold material phase.This s;'rnbol of passage lends significance to Noah's rain of forty days and nights; it is also reflected in the life of Moses, whose 120 y"lrs etr"o*passed three forty-year phasesand who waited

hamlet, town, -o,, 3, f, village, plur. settlement; city, communitn -",,t 3 i nu-t

The Eglptian glyph for "town" is an X in a circle, followed by a half circle that denotes the feminine gender Today we find the same glyptu the X in a circle, in street signs saying "No Standing." kr both examples "X marks the spot " this ground, this earth.

ggge. JlJUl Four basic "key" designs of classic architectural omarnent indicate the four ancient elements. Ape Factor. Mountaineers quantify the extent of their reach by the term "ape factor," the ratio of arm span to height. Since, on average, our height equals the distance of our outstretched arms, an ideal body fits within a square. Its diagonals cross at our geniials, indicating the square's relationship to fertility, btth, and generation. If arm span and height are equal, the ape factor is unity. A higher ape factor means a greater reac[ desirable for ihose who climb and hus the earth.



forty days on Mount Sinai to receive the Ten Comrnandments. The Israelites spent forty years wandering in the desert. Jesus'forty days in the wildemess, the forty days of Lent, and Ali Baba's forty thieves each recall the transformation of earth and self, often through physical ordeal. At the Iortieth day of human pregnancy, the embryo becomesa fetus.

THE BIRTH OFTHE SOUARE \A/hile the triangle was constructed by connecting existing points within tl::e z:esica piscis, *te construction of a square requires the faculty of rational thought: one goes through a sequential, logical procedure in order to discover the hidden points to cormect.(Some people are called "squares" and "blockheads" for being too rational.) There are many approaches to constrlrcting a square. Here are two useful ways: '

Earth's four-comered cracks. When you see three-cornered, 120' cracks in elastic materials, you know that the universe broke quickly there. But when you seefor:r-cornered crossings, 90ocracks in inelastic materials such as old paint, tree bark, porcelain sinks, tiles, and the inside of glazed ceramic mugs, you can be sule that they split slowly over years, decades, and centuries. Vly'hereas the molecular bonds in elastic materials continually adjust their strain in all directions, inelastic materials do not yield so easily, releasing their tension slowly in one direction. The longest cracks are the oldest. Four-comered cracks are the crow's feet of matter



The fust is the simplest and most elegant. It arises completeiy witlin a aesica piscls. (1) Constnrct a aesicapiscis.(2) Driw a hori_ zontal and vertical line connecting the centers of the circles and their intersections. (3) Put the cornpasspoint where the lines crossat the centerof the aesica piscis and the pencil at one circle,s center. (4) Tum a small circle. (5) Connect the four points where the circle crosses lines to make the tilted squarevisible. the




The second method is the classic one given in Euclid's Elementsfor constflTct'rrga squareupon a given side. the Fo11ow logical, rhYthmic dance of the compass with piscisto produce a lhe aesica square. Complete the arcs, turning ihem into full circles to seethe fuI1 dlmamic of forces.



TTIEWAVEOFTHE GODDESS Withour abilityto construct square canbringtheprina we ciples of the Tetrad to our universe. We can create volume and clothe it with the four elements, or states,of matter. The ancients s)'rnbolized this materialization process by weaving. The earliest known goddessin prehistoric Egypt was Neith, goddessof.weaving, wisdom, and war, later known to the Greeks as Athena. Although Neith is seen as a virgin goddess,her headdressof two crossedarrows symbolizes the polar crossings of male and female principles, of warp and weft, needed to make clotfu the ancient s]rnbol of matter. Diagonals represent crossing and fixing. Crossing, iike multiplying, represents fertilization; this is the origin of the use of an "X" to indicate multiplication. Today,we speakof "multiplying our progeny," "cross-pollinating" plants, and "cross-breeding"animals.The crossed arrows alsorepresent the diagonals of a square. Upon her four-quartered loom Neith weaves the four states of "mater" with which subsequent rnother goddessesclothe newbom forms. In the Greek alphabe! whose letters represented numbers, the words for "goddess" and "Gaia," or Mother Earth, eachadd up to fifteen, one unit from 16 (= 4 X 4), the square of a square and ultimate s)'rnbol of the archetypal feminine as fertile mother goddess; the ability to construct a square gives a geometer accessto this hidden framework. To explore the principles of the Tetrad, first we'll need to construct a square(by Euclid's method). Then quarter it by simply drawing and extending th e oesica piscis'svertical line




TffiX ffi ffiffiwM through the bottom and top of the square. Use this distance to walk the compassaround the square and find the middles of the four sides. Draw lines connecting these opposite midpoints to quarter the square.Also draw the square'sdiagonals. Diagonals are the means by which a square may be extended beyond itself or into its own depths. The use of * and X between the midpoints and corners aliows us to "harmonically decornpose" or "diminish" the square in a quatemary rhythm, giving us lines and crossings that maintain the "root-two" geometry of generation" as in a woven cloth. Connect the four midpoints to make a tilted square. Notice the points where the diagonals cross it, and connect them to make a smaller square within it. Now notice where the smaller square's sides cross the *, and draw lines cormecting these four points. Continue "weaving" this pattem, the ultimate s)'rnbol of material manifestatiory in as small a scale as the Doint of vour oencil will allow. Each squarecovershalf the areaof the prwious, larger square. Try doodling arid shading this design to see patterns emerge. Extend lines to see greater geometric detail. The pattern is a powerful mand aIa to gaze at. lr'Vhenits geometry has been properly applied by skilled artists who know the rules {or relating the sceneto the geometry, it can bestow a structure with a grrpping effect on viewers. Traditionally, only the images of deities align perfectly along the lines.

a Decomposing square.

Part of a Babylonianclay tablet showsthis construction astaught in the second millennium s.c.



"Worldly" characters are slightly " off ' the lines to indicate that the world's material forms are only approximations of ideal, eternal archetypes. This placement gives the images of deities a feeling of serenity and "otherworldliness," while forms placed away from lines have a feeling of motion and Life,of tirne and transitoriness. This was once part of a wellknown design canon based on nature and the science of visual perception. It is periodically "rediscovered," used, and shr.rured throughout history. At its best, geometry gives the artist great freedom. But bver tirne its use becomes mechanical and the art loses its spontaneity. Then artists shun geometry in favor of "free-form," each age finding its own expressions for the archetlpes *rat motivate it. Discover the "Goddessb Weove" in Noturol Forms A square with crossings has been constructed around both of these natural forrns. One has been "harmonically decomposed." Do the same with the other to see how nature's designs express this geometty in their details.

Harmonically decomposed square,

Tungsten atoms magnified two million times resemble watery ripples in a square pattem of energy vibration.



The skeletal design of a microscopic radiolarian follows the "lines of force" within a decomposed square.

eo I nspiration, enp assion, necessary is indeed for art, butthe creatioe of of knowledge theScience of Space, theTheory of Proportions, from far the narrowing creatiue of power theartist,opens for him an infinite aariety of choices within therealm o of symphonic compsition. -Matila Ghyka (Romanian royalty, educator, and wdter)

Discover the "Goddessb Weove" in Goddess Art For over fiJfy centuries, until recently, altists, craftspeople, and architects have consciously used the language of geometric symbolism as an r.rnderlying framework in their creations. They used different shapesfor different themes. The decomposedsquarewas often used to depict a goddessin her fertile, birthing, or nourishing character.Many examples from worldwide cultures use this same geometric s).'rnbolism, which arises spontaneously from everyone/s own archetypal depths. Gods and goddesseswere personifications of natural processesand principles at work in the cosmos and ourselves. Each of the following figures is enclosed by a square with crossing lines within it. Some have already been anaIyzed. Harmonically decompose the square around the others to seehow each was designed. As you do the geometry and see the grid crystallize, you'll see its relationship with the scene unfold and r4esh. Use what you leam to design your own compositions.

Tn, oi* of artisto represent theoutward not appearancethings,but of theirinward significance. -Aristotle


Arching with misplaced arms over the waters of chaos from which the cosrnic order emerged, the Egyptian sky goddess and mother of the gods, Nut swallows the sun at dusk amid the stars. She gives birth to it as dawn where its rays shine upon the face of the nurturing goddess Hathor on the horizon. (Painted ceiling relief, Temple of Hathor, Dendera, Egypt.)

Eg;rptian mother goddess Aset (lcrown in Greek and English as "Isis") holds the gl1ph for "etemity" while kneeling upon the glyphs for gold and perfedion. The scene'sgeometric center is just in front of her womb. (Sarcophagusof Tuthmosis III, c. 1450B.c.)

h my'th,the Egyptianvulture goddess Mut ("mother") was a virgin fertilized by the north wind. (Pendant from the tomb of Tut-Ankh-Amon.)



is I hebeautit'ula ofthe manifestation secret law:s nature. . . When of nature beginsto reaealher opensecretto a person,he feels an irresistiblelonging for her mostworthy art. interpreter, -JohannWolfgang Goethe (1749-1832,German poet) Sumerian pottery of the 5th

shouldl [P ainters fix truth eyes perfect on their of asa perpetual standard reprence, be to contemplated the with care, before they minutest proceed dealwith to canons about earthly thingsbeautifuL -Plato

lady of theBusts. (Detailof plate/Rhodes.)



Mater Mdkes Motter . . . and theworld wasvoid andwithout form and darkness was on the faceof the deep. *Bible (Cenesis) created shares the urge to create. Through the If you would understand miracle of birth the great cosmic creatingprocessexpresses theinoisible, IookcarfuIly its archetypes and becomes more conscious of itself. By at theoisible. bringing into expression the visidrlLocked in our hearts we resemblethat processand grow in selFagareness. Through geometric replication of nature's formations we can learn about the archetypal principles inherent in urriversal design and in ourselves. In ancient mathematicalphilosophy, the first stirring of the Monad resultsin the polar Dyad. Its geometricmetaphor is the construction of the oesicapiscis. In myth, it is representedas the meeting of primeval parents. A11 creation is due to polarity. Everv birth occurs Nr*brrc arethesources through this interpenetration;f po"ilti.'" urrd rrugutive, light ofform andenergy the and dark, male and female,god and goddess,electricityand in magnetism, and the two circles of the oesica pisciswherc geoworld. metric forms configure. \fhile modem science understands the universe as the -Theon of Smyrna (Second synchronization of electronic power with magnetic fields, centuy A.D.,Platonist vibratory pattems of electromagnetic frequencies, and mathematician) energy bonds, the ancient bards sang of the union of the Great Father and Great Mother. They saw the bright lightning of the sky galvanize the dark caverns of the earth into activity, namely, fertilization of the land with nitrogen. Temples, mounds, and caims around the world served as technological devices fertilizing the communities in which they were built. The ancients personified their vision of the subt-11 tle forces of nature in myth. Where modem scienceobserves Ihe distribution energy the triad of light, energy, and mass as E=mc2, of the ancients mythologized a mystic marriage and birth in three stages:a follows definitepaths 'u.thich field or womb of light arises, swirls as un ene.gypatte-, it may bestudiedby and physical forms precipitate upon the patiem. The means geoffietric of geometer replicates this cosmic configurating process with construction. three tools. lVhen the precise union of circlei occurs. the three-stage process of birth begins, cu.lminating in forms -Samuel Colman clothed by the four states of matter. -Talmud T




-l*.._.r (',


> .,,) / -

A Geometric manifestation from light tlrrough energy to matter' into sohererepresentingundifferentiatedliSht stirs, Polarjzing svmbotlied by two intersectingcircles'The two poles o'oposites swirl in oppositedirections in mathematicallypreciserhythmic by puti"*t, represented spiraling cones The sphere "n".qy u whirling energy field displaying lines of force upon b"cJ,tt"s whieh atoms configure to manifest material visible volumes'

The womb of the World Mother stirs. Electronicpower galvanizes magnetic energy into a state of activity, brlt there are no distinguishing {eatures. The geometer uses the compiscis oass to draw the womb as a circle around the oesica been referred to as the irom its center. In the past it has Monad, Orphic Egg, PneumaticOvum, and Individualized Field. Archetypal pattems arise. The electromagnetic field spins tike a sun on its axis. Masslesslight whirls in mathematically precise rhythms and pattems' This is Plato's "world of pure knowledge, the realm of ideas" and the "world of wonder" describedby seers.The Eglptians personified this field as the work of the architect/builder god Ptah. The Greeks called it Archeus, the builder of "first principles" (architect, arch, and archetype all derive from the

Orphic Egg surrounded by a spiraling snake, s).rnbols of the stirring Monad, the womb, or field of light giving birth to our divine Self.

80 -lI he worlds originate so


Creek word arche)accordrngto the will of Zeus,In Hindu mym, the creativewill oI Indra was carried out bv Vish_ thattruth maycome and vakarma, architect of the gods. The Freemasons .rif li U,r" dwelltherein. brancl Architectand Builder of the Universe.In Chrisfiar symbolism it is called the Mystical Light of the Logor- foifl" -Buddha geometer's eyes, the ueskapiscisinherently hold! the pro_ pornons and pattems of the basic shapeseven before they are constructed. goddessgivesbirth. The geometermakesvisible the .The archetypalproportions inherent in the aesica pisris, heloinr Ceometryexisted before rhe birth portal. In the electromagn"ti" fi"f J] :,hemihrg.ugh the spiraling, masslessphotons thicken into Hnl opposite_ thecreation. spinning whorls of protons and electrons that'braii with -Plato neutrons into massive atoms. The precipitating atoms are tied in Borromeanknots along the ircheiypal lires of force like threadsin a carpetor beadi on u too-,iit" -o.rrirrg Ju; on previouslyunseen spiderwebs,and like iron powde'r gal_ vanlzng nearan invisiblemagneticfield. Atoms,the blsis for the elements,precipitate in precisegeometric patterns, like lace embroidery or lslamic tiles, aiong these lines of force.The geometriesof nature,sfo.-s ,e.reil the pattern of Whot notur,,reates has their energy fields. Natural structures,then, are the arche_ typal matrix made visible. As ideal geometricpa tternsmate_ etemity it. in rialize they're clothed with four thi&nesses of garmen, four -Isaac Bashevis Singer densities of "mother substance,,'unfolding "as lishi and (190rt-1991, American thickening tfuough the phasesof gas, liquij, and iolid. ln author) ,:]Tt the Demiurge is said to ionqr". chaos by Tllll. dlviding it into the four elements. We can perceive the invisible lines of force by looking at . the of _geolneay nature's forms. For example, dissolrre"as much salt into a bowl of hot water as it wiliallow and let it The right thing is to evaporateslowly over a period ofdays. A cubic lattice of salt proceed ffom second crystals wilJprecipitate thismatrixflu id. The morestowlv in dimension third, which to the crystalsgrow, the larger they will be.Rapid evaporatioi brings us, I suppose, produces smail crystalsas seenin Hawaii,s volcanic obsid_ to ian stone,where molten lava hits water and cools immedi_ cubes otherthreeand ately. The crystals are a picture in our consciousness made dimensional figures. Dyour stowty perceiving nervoussystem,which we identify or project onto the geometricenergy_con -Plato figuration. Nature's pattems are basedonifie mati'ematics of three_ ,. qrmensronal space. In essence, nature,s creating process



yields its fruit by giving birth to volume. Constructing the universe, therefore, involves the process of volumization. The archetypal field where the patterns first appear is a sphere, a Monad. The geometry that arises within this sphere is obliged to manifest the Monad's principle of equality in all directions. Nature adheres to this principle by configurating primary volumes that divide the sphere equally in all directions. An experiment with a rubber ball and a piece of chalk will show that there are very few ways to put points aror.rnda sphere so that when they are connected by Iines they form identically shaped surfaces, edge lengths, and angles. Nature's first expressions in three dimensions are such forms that fit perfectly within the sphere and present us with an identical view in a1ldirections, no matter how you turn it. Theseforms, or "volumes," arebased on surfaces with the same shape, either square, triangle, or pentagon. Ruled by the Monad, three-dimensional nature is structurally disposed to the geometry of equality in all directions. There are only five volumes that fulfill this requirement of equality by repeating the identical comer angles, edge Iengths, and surface shapes around a sphere. Although scoresof stone models of each of the five types dating from 1500 B.c. have been found in Scotland,no one has ascertained their pupose. The five volumes.were describedby Plato in Timaeus and so are known to mathematicians as the "Platonic solids" and more forrnally the "regular polyhedra" ("many bases"). Four were identified by Plato with the four ancient elements, or states of mattet and the fifth, the quintessence("fifth being"), representstheir all-encompassing "cosmos." Their names derive fiom the number of faces they have.

"Heaven above."

The four "elements" of nature below.





Puaro "Heaven" Fire Air Water Earth


SsapeorFacs Pentagon Triangle Triangle Triangle Square

Faces T2 8 20 6

EDGs 30 6 12 30 12

ConNens 20 6 12 R

Dodecahedron Tetrahedron Octahedron Icosahedron Hexahedron (cube)



Mathematicians have long known there can be only five possible "equal divisions" of three-dimensional space. Highly respected, discussed, and wondered at in ancient times, the construction and study of these five volumes were considered the ultimate goai toward which the elementary constructions built, the culmination and pinnacle of ancient geometric and esoteric knowledge. Their construction coma prised the final books of Euclid's Elements, compendium assembled and arranged the totality of Greek geometry that without the original esoteric syrnbolism of the mathematicaphilosophers. His book began with the point, line, and plane and ended thirteen chapters later with the construction of these five volumes. The first thing one might notice about theseforms is that their faces are either triangles, squares, or pentagons. No other flat shaoe used alone will enclose a volume without gaps. These are the most economicalexpressionsof space from which nature's designs derive their characteristics. - Variations on these five volumes in nature are virtually endless. They're the basis for all crystals, the orderly repeatedarrangementof atoms. The first manifestations of the universe are geometic. From hydrogen to uranium all ninety-two natural atoms of the periodic table that compose minerals and crystals are geometric. At the borderline between nonliving forms and living creatures are the cold, herpes, and AIDS viruses, which are not alive but act as if they were and also take form



as Platonic volumes' Many microscopic life-forms share th qeometrv of these basic volumes. Each form's geometric Jxoressionis perfectly suited to its needsand purpose' Recently,a golf-ball m anufacturer discovered that by arranging tire bill's "dimples" into a dodecahedral pattern' he coild"avold having a seam around the ball's equator that slows and rotates it; thus, it can be made mote aerodynamlcallv stableand more accurate' 'The danceteacherRudolf Labanproposeda grammar of bodv movementsbasedon the relationshipbetweenthe Proportions of the human body and the five Platonjc volumes' 'He showed, for example, that the maximum angles through which human limbs move are identical to those angles of the the shape that gives the most twenw-sided icosahed.ron, syrnmetrical distribution of points, edges,and surfaceson the sphere.Thus Laban created danceswhose architectue in tim&as identical with the spatial harmony of ciystals lf a dancer had small lights on her limbs and danced in a dark room, the different Flatonic volumes could be traced by her movements. Many books have been written about these fascinating volumes. Particularly interesting are the relatiorships between them. For example, the corners of an octahedron fit in a cube and touch the center of each square face. Similar "duals" occur where the nurnber of comets of one volume eouals the number of faces of another. As this illustration an sh-ows, octahedral crystai can gradually grow into a cubic shape, explaining the endless variety of crystals.

THE PLATONICVOLUMES CONSTRUCT To read about the Platonic volumes is not enough. Constructing them is simple. In building these solids the georneter replicates the precise mathematical proportions of the archetypal energy rhyttrms that nature configures as the varPlatonic volumes are related ious rnaterial forms. Three;dimensional spacecanbe divided to each other. Lr this casean equally in all directions in only five ways by repeating one octahedron trarsforms into a and surface shape. The cube, represent- cube. comer angle, edge, ing earth, is the only volume built entirely of squares.Three volumes have triancular faces and are known as deltahedra




("triangle-faced" ). The dodecahedron has pentagonal faces and will be discussed and constructed in chaptet five. After constructing the models, hold and tum them in your hands. Get to know each volume by feel without geometrizes. looking. eaer The Birth of the Cube The cube, or "hexahedron," is the best known of the Platonic volurnes, the shape we know as six-sided dice but also found as salt and other crystals, as well as microscopic lifeforms. Geometers usually construct the cube either by folding paper or creating individual faces. Both produce fascinating results.

Salt crystal. Sodium chloride (NaCl) atoms tightly pack along cubic lines of force.

PaperJolding method: Construct six circles, intersecting pisces, show'r..Dtaw aesicas as straight lines through the circles' centers (seeillustration)/ extending some lines to reach the circumferences. Draw tab flaps where indicated. Cut along the lines and around tabs. Creaseand fold ihe lines and tabs to make a threedimensional cube. Glue or tape the tabs to the sides they encounter.



pisck. Cut out the Facemethod: Bring a square through a ztesim circle around the square, and use it as a template for making six. Cut a slit along half of eachsideof eachsquare. Slide the circles together through the slits to build the cube. You may wish to glue the rounded edges together to make the cube firm. Notice how the cube formed by the lines is inscribed or encasedwithin the curved seamsof a sphere.

He is inzitedtogreat thingswhoreceiaes small things grently. -Flavius (490-583,Roman legal writer and politician)

Make a cubic soap bubble. Bend a wire into a cube with a handle. Dip it into soap soiution and let nature herself show you the minimal lines of force upon which she builds forms. If you capture a single bubble within it, a curved inner cube configures.



The Birth of the Three Dettohedro: Ttrohedron' Octohedron, ond lcosohedron Deltahedrareferc to the three Platonic volumes whose faces are equilateral triangles. The paper-folding method is pisces,and conshown first. Construct intersecting aesicas triangles equal to the numnect points as shown to produce ber of faces (four, eight, or twenty). Cut around the tabs, fold, and glue. This models the ideal geometry that the Many, the microbes, crystals, and molecules, aPproximate'

s@ Tetrahedral forms Octahedral


a7 IT

I ruth is soex.cellent, that if it praises small but thingstheybecome noble. -Leonardo da Vinci

Icosahedral forms

Viruses associatedwith the common cold, herpes, measles,HIV take the forms of Platonic volumes.

I thinkthot-odrrn physics definitely has decided faaorofPlato. in In fact thesmallest unitsof matter notphysical are objects theordinary in sense; are they forms,ideas whichcanbeexpressed unambiguously in only mathemati Ian age. cal gu -Wemer Heisenberg (1901-1976, German physicist)



The next method involves building solely by repeating the constmction of a triangle inscribed within a circle' To make it, first construct a large triangle within the circles that createa aesicapiscis'Ftnd the center of this triangle by connecting each oi its corners with the midpoint of its opposite side. P:utthe compass at the central point where they cross/ and open the peniil to one comer of the triangle' Swing a circ1earound the triangle. Cut around the circle. Create one as a tut out four to construct tetrahedron, a templateto trace. build an icosahedron' eight ior an octahedroryand twenty to Cut a slit along half of eachside of each triangle as shownLink them to configure the three volumes. Glue the curved flanges together. Notice how the volumes seem inscribed in spheres.

lThe Creeksl foundin the of fioe construction the in bodies one Platonic of the sphere, picture the the unionof oryosites, the workof theDemiurge, artistof theworld. crcatiae -Frederick Macody Lund




. After building them, hold them, tum them, and look at them. They offer both an intuitive and inteilectual education in harmony. First, get to know thern by touch without looking at them. Then, for intellectual interest, you can verify the discovery of eighteenth-century Swiss mathematician Leonhard Euler that in each of these five volumes, the numberof plus thenumber faces always morethanthenumis corners of two ber of edges. The numbers involved in these volumes have been for:nd to be identical to the spatial representations of musical harmonies. Crystals, then, may be considerecfrozen music holding the proportions of musical intervals in the relationships of their comers/ edges, and faces. The forms of nature configure as visual music, "Crystalloidal tectonics" is a term used to describe nature's rhythrnic architectural designsin tluee dimensions. The simple beauty of a crystal is, in part, its strict mathematical order. Through the study of number and shape we begin to apprehend cosmic order. Now we can seehow the world virgin is fertilized, swirls into archetypal pattems, gives birth as the world mother, and then nourishes and clothes the geometric pattems with substance, the four ancient "elements," manifesting as atorns in configured pattems we can sense.

The unhmseis the externalizatianthesoul. of -Ralph Waldo Emerson

FOUR CORNERSOF THE YOUNIVERSE A comlnon theme in worldwide rnythology is the human as microcosm, a miniature model of the whole universe. Now dismissedasprimitive, this view was long considereda key to the understanding of the Self. The ancient philosophers saw their irmer lives arranged according to nature's own harmonies, for in nature, geometry, and their spiritual lives they discovered the same principles. A fundamental map of ourselves is found in the mathematical intimacy between the Triad and Tetrad. The ancient mathematical philosophers saw themselves wherever three and four mingle. Three Borromean Rings intersect at four just as light's three primary colorsblend secondaryspaces, to produce four more. This simultaneity of three and four is a simple blueprint of ourselves: the triangle denoting a

Nr-brr...shouldnot beunderstood asa solely construction of consciousness,also but as an archetype thusasa and constituent nature of both without within. and -Marie-Louise von Franz



Tibetanproportions f or depicting the Buddha.

divine hinity "above," or deep within, and below it a square unfolding the four elements of familiar human nature and body outward. Trinity and quatemity is representedby the symbol of the pyramid on the Great Sealof the United States,whose divine eye in the triangle floats in light above its four-faced, This "three above and four square-based brick fourLdation. below" is also s).'rnbolized the Sphinx's trapezoidalheadin dress,which implies an unseentrinity aboveit (seechapter three).Our one SeIfhas an inward and outward expression. In painting and sculpture this theme was often depicted by delineating the head in a triangle and the body below within a square frame, as in this Tibetan plan of the proportions for correctly drawing the Buddha. (Geometrically decomposethe large squareto seeits inner design.) Note also the famous s)..rnbol of the Freemasons, the compass over the carpenter's rule. The compass is open sixty degrees,the angle of an equilateral triangie, above the ruIe's ninety degrees. The compassand squarerule together spnbolize our inseparabledivine and human natures. Few would suspectanything special about the design of doors, windows, and roofs. So common are weightsupporting triangular roofs and lintels over rectangular facades, doors, windows, and porticoes that we hardly notice them. But this is just the effect the enlightened archi-

Th, trtrod,o*prehends theprincipleof thesoulas weII as that of corporeality; for theysay that a lioing ffeatureis ensouled the in same uay that thewhole unioerse arranged is accordingto harmony. -Iamblichus A triangle above a square in art and architecture traditionally represents our deepest divine nature "over" the four "elements,, of our human nature.


tects o{ ancient tirnes wanted for embedding esoteric instruction aboi.rtthe self. The architecture itself, its proportion and omamentation, held the esoteric teachiag in plain sight, a reminder of their divinity to those initiated in the timeless wisdom that speaks of self-knowledge and spiritual transformation. Copies of ancient architecture are scattered tfuough our cities. The wisdom-teaching is still evident to those who notice and can read it. But this century's trend in urban architecture has been simply to hang walls on skyscraping steel frames. Triangular lintels are no longer used to support weight and pass along the traditional teachings that, until recently, had endured for thousands of years. By allowing the square and cube to eclipse the divine triangle frorn common sight, we've replaced a valuable body of knowledge with soullessarchitecture.

THE FOUR ELEMENTSWITHIN The ancient philosophers saw the natural world of earth, water, air, and fue within us as the quatrain of our soul. In this irurer symbolism, the four elements are seenas four levels ol motivation within ourselves,a human tetrachord upon which the forcesof nature play. The ancientssy'rnbol2edit many ways (see chart below) in architectwe, r"tigiott, mythology, and arrangementsof society.Sincethe whole earth was symbolized by the human rnicrocosm, it is zue who are referred to as the "four comers of the globe" containing the ,,sevenseas,, (chakras),as well as the four stagesof a spiritual journey and four levels of a ternple or socielz Adam's dominion over the earth (in Hebrew,adamameans earth") refersto our ownmas" tery over the four realms of our inner nafite, representedbv beastsof the earth, fish of the seas,birds of the air, and ,,creeping drings" (snakesand salamanders,kaditional sr.,rnbobbf fire). To r-rnderstand slrmbolismeachof us -rrt b".o-e this famfiar with the ecology of the world within ourseives. /hat is represented within us by the elehent ,,earth,,? We form identities from pictures of the world as they appear to our senses, sufaces and solid bodies.The earth nafure as within us relates to what is most dense,most slow to change, transform.

Fire Air Water Earth Seven"chakras,"or centers of motivation, along our spinein four levels:earth (pelvis),water @e11y), air (chest), tue (head). and




and I heweather my a moodhaoe little my I connection.haoe foggy and myfine days within me. -Blaise Pascal (1623-1.662, French mathematician and philosopher)

ETE The Hebrew letters YHWH (English: "Jehovah") when written vertically depict an upright person in four levels, image of the cosmos.


\ rytat"water" is within? The ancientssaw our emotions in the "moods" of water: surging, rising and falling, iurbulent, stormy/ calm, sometimeslaughing along a mountain stream or iaining in tearful torrents, our emotions as part of our inner hydraulic cycle. Like water, emotions take the form of their container. Our emotions are shaped by our essence by worldview, family traditions, opinions-in our thoughts. Thesethoughts Iorm the container that shapes If your emotions are in a particular habituaorr, "moiott.. the thoughts and beliefs that have created the pattem, find iigid pattem. Work to develop new thought pattems, and it won'i take long to seethe emotions take different forms. The "atmoiphere" within us is where our thoughts fly' Soaringflights of fancy,daydream castlesin the air, the rarefied intellect, and clouded thoughts all refer to the influenceof reason.The Egyptian artists symbolized this influenceby depicting birds, sometimesnetted, sometimesfree and soaring upward. Our thoughts, like the wind, are a pollinating force, spreading ideas over our inner landscape. -The element of "ffte" is represented within us by our intuitions. "A flash of insight " "it hit me like a bolt of lightning," and "seeing the light" are all phrases referring to a glirnpse of our deeper Self, the very power by which we are conscibus. Intuitions are most ephemeral, the most electric, the least dense, and the most difficult of the elements to discem. When we are rnost in touch with our deeper Self we are not forming picftrres, thinking, or emoting but are simply aware of our inner "fire" and senseof purpose, experiencing a deep knowing beyond thought or etnotion or sense.Fire is a symbol of purpose. The various ways these{our statesof matter have been s)rmbolized over the millennia as states within us are too nurnerous to explore in detail. hr general, they describe a fourfold structure of the world within, symbolized by the tetmgrammaton(Greek for "four letters"), the Hebrew letters YFIWH (in English, "Jehovah"). These correspond with the four castes of Hindu Vedic society (priesf warrior, merchant, and servant) that comprise the body of society in the image of Purusha, the Cosmic Man. Similar in design, the structure of medieval European society separated monk, knight, burgher, and peasant



Plantonic Volumes (Regular Polyhedrs) Tekahe&on Octahedron Icosahedron Hexahedron (Cube) Genesis: Adan's Dominion Creeping Things Birds of the Air Fishesof the Sea Beastsof the Field

Ancient 'lElements" Fire Air Water Earttr

Egyptian Animal Symbols CatlLion Hawk Crocodile Beetle

Egyptian Stone Symbolism


Granite(Igneous) Master Driver Alabaster Horses Limestone Carriage Sandstone

Biblical Symbols Pillars of Fire Whirlwinds Water Dust Ancient Temple Structure

foumey of the Israelites Promised Land Wildemess Red Sea Bondagein Egypt Rivers of Paradise Honey Mitk Wne Water

Transformation lntention Attention Concentration Meditation Life of jesus Resurrection Crucifixion Persecution Service

Somatic Divisions

Head Heart SolarPlexus Genital Human Body Systems Nervous Respiratory Circulatory Digestive What Do You Desire? (Buddhist) Liberation Merits Riches Pleasures

Four Ageb Gold Silver Bronze Iron

Holy of Holies kmer Chanrber IrmerCourt OuterCourt Hindu Castes (Bodyof the Universe) (Priest) Brahmin (Head) (King/Military) Kshatriya (Arms) (Merchant) Vaisya (Bel1y) (Laborers) Shudra (Feet)

Medieval Society Monk (Clergy) Knight (Military) Bugher (Commercial) Peasant (Agricultural)

Medieval Nature Spirits Salamanders Sylphs Undines Gnomes

What Do You Cling To? (Plato) Wisdom Knowledge Opinions Ignorance

Many quatemaries symbolize the world as four elements, or levels. They represent four levels or centers of gravity within us with which we identifv and expressourselves in the world. The purification of each level represents four stagesof transformation and transcendencetaught in myth and religion.



with the four humors and the four somatic roles associated divisions. Today,we might considerthe four blood types. Like the temple, mandala,and cathedral,ancientsociety was sffuctured as a model of the cosmos and man. The Egyptians, for example, saw the scarab beetle, like most other insects, undergo four stages of transformation from silent egg to voraciously hungry larva to cocoonedpupa to winged aduit. They used this process in their hierogllphics and mythology to sgnbolize spiritual transformation from a mundane life to initiation and the mummy-cocoon from which the priest-initiate emerged as a spiritually awakened adult. As is evident in the stone symbolism in their architecfure, the Egyptians were well aware of the four elements. In their modeling of the cosmos,"earth" was representedby sandstone,"wate{' by limestone (formed of compressed seashells),"air" by translucent alabaster,and "fire" in their stone pottery, statues, monuments, and temples by any igneous rock such as granite, basalt,and diorite. The ancient fourfold syrnbolism appears in many forms frorn architecfure to the origin of playing-card suits. It was intended to remind and teach the initiate about himself so he might, to the degree of his understanding, come to know himself as a model of the world. Carl Jr:ng recognized the fourfold nature of Hindu and Buddhist mandalas as symbolic of the Self and its joumey through the four elements, which he labeled the "four orienting functions": sense, emotion, thought, and intuition. As part of their analysis, he had his patients construct personal mandalas. The first three elements are easily available to ordinary consciousness.The fourth, our intuitive fire, is like the star followed by the Three Wise Men. When our three lower elements become purified of destructive habitual patterns, they become "wise" and follow the spiritual light above and within. The geometer who takes this study from its outer symbolic form to the inner sacred experience does so through meditation. Attention is directed not to any geometric construction but inward to one's own ecology. Self-discovery and self-knowiedge require nothing outside us, only sustained attentiondirectedto our irmer landscape. not easv It's





them and niix

world we have els of the

emotior$, or thoughts. We slip b6tween tossed about by winds and waves of the within. Focus on each of four levpeivis, gut, chest,head-and come to

know them as encouraging the concentrate on ancient p In ter is building

of earth, water, air, and fire. Avoid images.Just of pictures, scenes, Get to l<rtow your world as the feeling. did, as a representation of *re cosmos. the patterns of the universe the geome-

Ittnpr IR

Irv r

RegEltEra It is a frequent.assertion of ours that the whole universe is manifestly completed and enclosed by the Decad, and seededby the Monad, and it gains movement thanks to the Dyad and li{e thanks to the Pentad. ' -lamblichus I4trhere plants have five-fold pattems, a consideration of their souls is in place. -lohannes Kepler The mathematical rules of the universe are visible to men in the form of beauty. -]ohn Michell (1933-, Englishphilosopher, antiquarian, geometer, writu) . . . the irregular and vital beauty of the pentagram. -Claude Bragdon(1866-1946,Americanarchitect,philosopher, andwriter) Let us build altars to the Beautiful Necessity,which securesthat all is made of one piece. -Ralph WaldoEmerson There is a geometry of art as there is a geometry of life, and, as the Greeks had guessed,they happen to be the same. -Mtttila Ghyka


Birth of the Pentagon and Pentagram Star. 96



sixty rta bionsinclude a five-pointed star?Flagsaffectpeople oeepry. I here must be something very powerful in the star icon if so many natioxs with conflicting ideologies each claim this samesymboi astheir or+'n.One would exiect each country to design its flag to be individual, unique. iVhat is it they all vie for? Vexillologiststell us that the alpearance of star-spangied flags continues a ritual dating baik millennia, when flags were seen as magical slnnbol-s of power and invulnerability. Everyonewants to be associated-with excei_ lence;the star's appeal emergesin the ceremoniousrespecr accorded flags and five-part phenomena. We encounterthe stal paitern ceaselessly comrnercial in logos. Modem advertising agenciesspendbi ions analvz_ ing aestheticpreferences and have rediscoveredthe univer_ sal appeal of pentagonal s)'mmetry. Something inside us ctearlyresonates with the fivefold form. Justlook at the open embrace of a well-constructed pentagonal star and noice -world the feelings it evokes within you-.The recomizes and respectsthe five-pointed star for the same reasonsiit imparts to everyone an irresistible impression of excellence^and power. From ancient times this star has been a magical sr,rnbol fnr warding off evil in Babylon,Egypt, Greece, trdi", 6hi";, Atrlca, the Americas,and elsewhere. In literature, Goethe,s Faust drew a pentagram star to yard off the devif but as it was constricted"imperfectly, Mephistopheles was able to approach him. the'maeic;i association this star continueJtodayin military matteis of as a ranking system ("five-star generais,,)arnong *,ose, for example, who meet at the pentagon, tt" *o.fa,,"t office building, near Washingto.,,"D.C. Along lritL ;;?;l_ ,ul:u power the pentagonal star induceJ a feeling of Td autnonry/ rurther explaining its military pr"ser,ce ind appearance flags and in commerciallogos. on I44ren the star is inverted it,s called iDrud.enfuss (Ger man. for l'w_it$t foot,,) and appears in magical ,it"ui urra witchcraft. It has been ussociried with deviis *t d;;;;; sincethe rebel Egypfian god Setresistedspirit"uf urpirutloi the adversary.of his brother Osiris and nephew :: ".*T:\et Horus. had many attributes associated with the modem [nage of a de\,'11; horns, coarsered hair, hOOveS, tail, He a


"You can trust your car to the man who wears the star . . .,, (195!s TexacoOil Company Jmgle,




and his hordes resist truth, order, justice, and righteousness. His forty-two attributes are actually s).'rnbolsof the downreaching elements of our own psyche that must be purified. The Eglptian "underworld," the duat, whose hierogllph is a circle surrounded by a five-pointed star within a circle, is the mythical nighttirne place where the "perishable" stars go when they sink below the horizon at dusk. The duat recalls our own spiritual sieep from which we can awaken. The Egyptian star gJyphwitiout the circle signified either a nighttime sta1,a door, or a teaching, an inhiguing combination of ideas in one glyph. The sun's inevitable birth into dawn and daylight (s1'rnbolizing our own spiritual arising) is resisted in the duat by hideous adversaries (illusions within us). In ancient Greece, the star was associated with Pan ("AIl"), who alsohad homs, red hait hooves,and a tail, representing the lustful fertility of nature, which induces p_anic and,pandemonium ("a11 demons,,) in prim people. the Later in Westem culture Pan's sexuality became thi great temptation of the modem devil and the itar was invertEd to imply the reverse of, and resistanceto, its positive qualities of excellence and goodness. the pentagonal star has been a powerful psy. Sl*"I-y, chological ico,nin the past and present and undoubtedlyiill be so in the future. It conveys an impression of exceiience, authority, and uprightness and strikes us all with a power and strength that can only come from the deep ievel of archetypes within us. _ But exactly what is it about the five-pointed star, and not other geometric shapes,that appeals to us in this way? What is the source of this star's power and excellence? From where does the star derive its authority? I4lhat are the spe_ cific principles of the Pentad that touch us so deeplv? Wirat is the message this star? of

T o hold,as'twere, the

TTIEWORLD'S OLDEST LIVINGSTAR Fivefold symmetry.fills_a special place in nature,s design Ie[con. lne nve,pomted star's associationwith excellence, power, and authority is manifested in nature as beautiful and efficienfly designed forms. pan,s fertility was svmbol_ ized by the five-pointed star becausethe shipe is a kev to

mirror tonature. up -William Shakespeare



nature/s fecundity and generative powers, providing- the

iiself as living forms and in the attributes ot nessexpresses living creatures. It was for this reason that the pentagram star *as a s)'rnbol of humanity and health to the Pythagoreans, who wrote the name of the goddess Hygeia around its points. Pentagonal s;rmmetry is the flag of 1ife. When it ippears in nature we are seeing the archetype of the Pentad expressiflg life's excellenceand authority. We can't help but encounter stars and pentagonal structure throughout nature. The overall design of any leaf fits within a stretched or compressedpentagon. Leaveshave the samesymmetry as hands. Look at any bush or tree and visualize its leaves as open hands catching sunlight'

of and fullness, fertilify of life.Thearcherype fivefreshness,

is The Pentad e particularlycomprehensia phenomena of thenalural of theuniaerse. -lambfcnus

Leaves configure their archetype as stretched or compressed pentagons. The One urrderlies the Many.



d C@F Apple Pear Blueberry Carambola

Apple slice

Seacucumber (crosssection)

Sand dollar (seaurchin)

Microscopic radiolarian skeleton

Pentagons abound in the animal kingdom. The phylum Echinodermata, represented by the starfishes, seacucumbers, ald sand dollars shown here, is particularly obvious in its pentagonal symmetry.

Or look at the bottom of anv familiar fruit to seethe nentagonal remnant of its five-petaled flower. The flowdr of every edible fruit has five petals (though not all five-petaled flowers yield edible fruit). Before you eat an apple or pear, cut it in half at its equator to seethe star paftem in which its seeds,the holders of life itself, are arranged. When we look at any IeaN flower, or fruit we're seeing the invisible energy web of the archetype made visible as i pattem of living cells.

...thoucunning'st patternof excelling nature. A STAR IS REBORN -William Shakespeare

We can gain insight into the Pentad's exceilenceby geometrically constructing its representatiory the pentagon. [r this shape, its most elementary visual form, we'll diicover spe-



The Roman numeral representing five, sy'rnbolized by the letter V, derives from the shapeof the space between the open thumb and fingers. The Roman numeral for ten, the letter X, is actually two V's.

The human body clearly expresses Pentad'ssymthe metry in its five sensesand five extensions from the torso, each limb ending in five fingers or toes.

o Pentadactylism, the fivefold symmetry of hands,paws, and claws, expressesthe Pentad, life.

cial inner relationships that will help explain the power of the star and its archetype. A pentagonis among the more complexconstructionsto be brought through the aesfua piscis.But it needn't be difficult. Of the many ways to constructa pentagon,the sirnplest occursevery time we tie our shoelaces a bow tie. The inior tial knot producesan internal "good-luck" pentagon.To see it, tie a strip of paper into an ordinary knot. Tighten it carefully and press it flat as you gently pull. Hold it up to the light to see its inner pentagram star. Tie five such knots at regular intervals along a long strip of paper to make a pentagonalframe with a star at eachcomer. Another method is to start with a paper square representing the "four statesoI mater."Fold and cut it as shown in the illustration to produce stars in positive and negative sDace. Construction using the geometer'sthree tools requires more complex steps than we experiencedin consffuc[ing a triangle or square, as well as a deeper knowledge of the relationships within Ihe aesica piscis .Few would discoverit acci-

Tying a pentagonal knot.





Classic puzzle: How can you plant ten flowers in five rows and have exactly four flowers in each row?

r'ffiry .s\ tr = | >k 2\


relsruereluade;os'ur A <v 7)


| .tkz^l


dentally or by trial and error. The oldest construction on paper was taught by Euclid in hjs Elemmts (c. 300 r.c.). although much older Babylonianclay tabletsdescribingthe steps have been unearthed. Periodically throughout history the construction of the pentagon has been so revered that it was kept secretby var_ ious. societies who recognized its predominance in. nature and its potent effecton human natuie. They studied the.pen_ tad's principles in geometry and nature,and its psycholoe_ ical effect,deeply enough to seein its application t ,o*'i_ edge that could be misused.Around 50d i.c. the pentagram " star was the sign used by advanced members oi the rythagorean society to recognize one another, and one thou_ san_cl years later it was a guarded teachingtool only taughr orau_y, written about,by the craftsguiJdswho in?used*its not symbolism into the designs of Gothic cathedrals. It wasn,t until 1509when the monk Fra Lucapacioli, inventor of dou_ ble-entry bookkeeping and the mathematics teacher of Leonardoda Vinci, published De Dioina proportione, the that method of its constructionand unique geometricproperties was_revealed artists and philosophG in public. to out in the open again. Use tire geometer,stools, ard . ,,It's tortow these steps (seeillustrations on the next pages) ' to construct the pentagon and pattems that come from It. Some constructions of the pentagon you may come acrossare actualJyonly approximate, althoueh very close, but this method producesone that is perfectly froportioned.



Construct the pentagon. Follow these steps with the greatest precision and the sharpest pencil point.

itself, encompassing 1. Construct a z:esica piscis. and interpenetratlarge Draw a line connect- ing the four states of matter. ing the circles' centers and a vetical 3. Using the sarne line up the middle. compass opening, put its point at the 2. Put the point of the compass where center of the large the lines cross, and circle, and turn open it to the center another small circle, of one of the circles. making a small Turn a circle within aesica piscis. piscisas tl,:'e aesica you did to consuuct 4. Draw a vertical a square. The penta- line up the small gon represents the piscis.Find its oesica "fifth essence,"life center where it

crossesthe line betweencenters.

6. Put the compass point atop the small circle, and open the pencil to the point 5. Put the unmoving leg of your com- created on the circle's diameter. pass at the center of pisSwing the compass oesica Ihe small upward until it cis, and open its crossesthe small pencil to the point atop the small circle circle. This new point and the point within the large atop the circle are aesica piscis.Swing the pentagon's first the compass downward until it crosses two comers.We the horizontal diam- have organized Monad by crystaleter (or complete Lizing the Pentad the arc to the full within it. circle).



7. Using the same oPening, compass walk the compass around the circle to find each new Point until all five Points are identified. Your pencil's final steP should end precisely at the toP where it began. (Helptul Tip: This construction requiresgreat preci sion. If your com-

pass was off even slightly at any stage, the difference will be magnified and the pentagon loPsided. To double' check, walk the compass around the circle in the opPosite direction. If there's any differencebetween the marks made in each directiory the actual points of the penta-

gon are found halfway between them.) 8. Connect the five points to make the pentagon visible. Welcomethe emissary of living structures,the geometric flag of life itself. 9. Corurect alternating points to

reveal the pentagram star. 10. A varietY of pentagonal forms can be constructed by this method, eachof which the expresses Penprinciples. tad's Practice constructing thern. Save one as a template for producing others quickly

@ o o @f f i * *



BUILD A DODECAHEDRON (Greek for "two The fifth Platonic volurne is the dodecahedron plus ten faces"). Like the four other volumes, it expressesthe principle of equality in all directions, having twelve identical regular pentagonal faces,thirty equal edges, and twenty vertices rneeting at three-comer joints. It is closestof the five forms to having the volume of the Monad's perfect sphere. As "quintessence" it isn't aboaet}jrefour elements, the configured statesof matter, but encompasses them, infusing the force of life and excellencethroughout their structures. Even nonliving crystalline minerals take part in the cycles of life when they're ingested by plants, animals, and humans, thereby becoming incorporated into our bodies. Like the dodecahedrory life's cycles encompassall the universe in its processes. Today, we encciunter dodecahedra, if at all, as those twelve-month desk-calendar paperweights and as special dice. The ancient mathematical philosophers revered the dodecahedron for its geometric properties and s).'rnbolism. Like a pomegranate ready to burst forth its seeds, the dodecahedron representsthe archetype of life and fecundity made visible. It's a three-dimensional pentagonal web on which life expressesits fullness. The best way to appreciatea dodecahedronis to build one. Unlike triangles, squares, and hexagons, pentagons cannot fill flat spacewithout leaving gaps. But twelve pentagons will enclosethree-dimensional space as the Quintessenceencloses elements. the



Sorrotrcsaidthat,from like aboTre, Earthlooks the oneof those tweloe-patched luthernballs.

Scientists have recently constructed dodecahedral molecules using carbon and various metal atoms that join to form larger molecules. Recently, a third form of pure carbon beyond graphite and diamond was discovered and formally named "Buckminsterfullerenes," or 'lBuckyballs," after the mathematiciary architect inventor, and visionary R. Buckminster Fuller, who used similar shapesfor his geodesic domes. A Buckyball consists of sixty carbon atoms,fiae arowrd each corner of a dodecahedron.

Dodecahedral skeletons of microscopic radiolaria.

r09 Hold and tum the dodecahedron to appreciate the beauty of its lines and surfaces. Get quiet within yourself and be aware of feelings this fifth Platonic volume evokes.

WHOLES IN NATURE'SFABRIC Avoid extremes-keep the Golden Mean.

-lleobulus Lindus cmtury one theoSeam (sixth of B.c., of ,,y;6ffi

Nature expressesthe cosmic phitoroptry tt geometry. "orrgh take us A deep look into the star's unique geometry will even closer to the core of the archetype of the Pentad. There, perhaps, we can discover the source of its power and the secretof its appearancein nature and its psychological effect on us. A clue is found in the ability of vegetation and certain creafures to regenerate another whole like themselves from a part, not only from their seeds. For example, when put near soil, the leaves of some plants send out roots and grow whole new plants. Flatworms and other creatures with prirnitive nervous systems also regenerate new creatures from their parts. Most familiar is the ability of starfish to regrow a lost leg, while the lost leg grows a whole new body. We can seethe principle of regeneratiory the whole in the part, in many places. The veins in a leaf reveal the branching pattem of the whole leafless tree. Each ridge of a fem's leaf models at once the whole leaf and the whole plant. Each seedon a dandelion'shead mimics the whole head in miniature. Il/ho hasn't noticed that the tips of broccoli and cauliflower resembleminiature trees? Look at the top of a fulI broccoli or cauliflower head while squinting and you can see the overall pentagonal groupings. Five major clumps contain constellations of smaller and smaller pentagonal structu-res:each clump is made of five smaller clumps, and so on. This is the rhythm by which vegetation grows, a pulsing fivefold rhythm. Each successively smaller branch models the others and the whole. This whole-in-the-part principle is the basis of bonsai, the Japanese of growing miniature trees and bushes from art the branches of a large tree. A tree's leaves swirl around their

The veins of a leaf, dandelion seeds,and branches of broccoli model the whole plant in eachpart.


twies in the samepattern that twigs swirl around branches' aroun-dthe boughs, asboughs emergefrom the asb"ranches tmnk. Each taken by iiself is a model of the whole' The next time you can dwell with a bush, look for this structural selfsyrtttttetry. Distinguish each level, and build your vision l-Ut yori can seJ a[ the structure comprehended as the whole. Cognizance of this harmony in nature and rnathematics attunes us to harmony at our owrl core. Today,we limit the word "symmetry" to mean a simPle rnirror image, such as our two hands. But the ancient Greek word symmetriallterallymeant "alike measure,"a term pelfectly zuited to these expressions of self-similarity on different scales. This fascinating characteristic of living forms curiously corresponds with the geometric property of the PenBFam to repl-icateforever smaller and larger versions of itself' The star ieplicates and inverts itself in its own central pentagory whictr in tum has its central Pentago& in an endlessly diminishing series of inverting stars. Flip the centrai pentagram outward and it transforms at each of the star's five points into a whole pentagram' These become nuclei for smaller stars to replicate within, in its center and around its points, multiplying endlessly. The Pentad holds the principles of the geometry of regeneration' It is


nature's way of producing endless variety using a single self-reproducingscheme. Pentagonsperpetuate their own image in endless detail. We are dranm into them as into the eyes of a lover, as we see the whole by looking at the part, Know one star and know the whole. This self-similar accord, repeating the same shape on different scales,is the basis of {ractal mathematics, which is at the heart of modern chaostheory. For example, the microscopic bumps on a grain of sand at the seashoreresemble an aerial view of the whole beach'sshoreline.Each small part of a turbulent system tums out to be a model of the whole and can be described mathematically by self-replicating formulas. Even in chaos we find the signature of cosmos. This part-as-wholeoccursin thelcientific art of producing holograms. Unlike ordinary photographic negatives, in which a piece from the comer holds only a small part of the picture, any piece cut from a hologram contains the information of the whole scenein miniafure. Explore the Pentogonb Regenerotion Construct a pentagon or trace the five comers of any illustration, and connect them to enclose a pentagram star. Then do theseactivities: Doodle pentagonsand pentagrams.Explore pentagons on your own. Draw them freehand, with each hand, as a continuous line. Extend lines and

The pentagram exhibits selfstnilarity in endless variations. Explore its regenerating geometry on your own by subdividing each part into smaller pentagrams.




I belieae geometrit the proportionseroed the Crmtor asan idm whenHe introduc d thecontinuo s e u generationof similar objects from similar objects. -Johannes Kepler

connect points to seehow the star regeneratesitself endlessly into smaller and larger moiels of the whole. The pentagon is extreriely rich in enJlessly lascinating self-replicatinggeomefry that is accessi_ o1" y,rh u.."13.ppencil point. Discover its galaxies ot stars, all identical in shape, differing on[z in size. 1. Decomposeand fold a penragon. 2. Geometricallyconstructa pentagonand connect rrs comers to inscribea pentagram star. 3. Connect the comers of its central pentagon to draw a smaller but inverted pentagram. 4. Extend the lines of this inner pentagram out_ * they meet the sides'of the"large pentaI^1-.9 *r, gon (seeiltustration on page 111). Use this as a tem_ plate to makeotherpentagrams. Use the straigLrtedge crease to sharply all *re ,. rmes both ways. Feeland experiment *iih the p"n_ tagon's tendency to fold various ways. Get all ire points to_meet the top and let them sprine at Ua;k; thls shou.ldremind you of a flower bud ope-ning. the star tnto pieces.Construct another sJar, ,Lut and cut the whole into piecesalong the lines. Re_ arrangethem to make smaller and larger starsand pentagons. Notice their interrelationsiips. parts added together match exactly with larger ones. . Construct dodecahedra. ilepeat the"corrst u"-

Intuition is theclear conception thewholeat of once, -Johann Lavater (1741,-1801, >wss poet, mystic, and philosopher)

li:1, 1"::p::ys srdeof eachof the small outer triangles(see itius_ potathesidesipward, puUt113) anJ

apentason. u srituro.,jrn" cut

Iif,"" 9" strp.each small triangle under (or over) the piece next to it. Clue or tape it there as a tab, corurectine me whole into a ,,basket,, resemblingan open flower. Two of_these basketsjoined fice_to'_face will rorm a dodecahedron. twelve of these Or baskets can be joined back-to-backby their sides to _rke u targerdodecahedron. With a needleand thread vou can sew stuaightlines in three dimensions betwJen


the points and comer folds of the pentagonal "basket" to reveal unexpected and beautiful geometric relationships within it.

m r\,,,n r-v/t\-4


Discover Pentogonot Setf-SUmfnetrg in Noturol Forms Each life form below is enclosed within a pentagon. Use a straightedge to connect its five corners to inscribe a pentagram star. Further decompose the pentagons and notice the remarkable detail with which the creature exDressesits archetypaldesign,and how eachsmall part is structured as the whole. Some examples have been completed to different degreesto show you how it's done.




Starfish mouth




RIDDLE THERABBIT It is worihwhile for anyoneto havebehind him of a few generations honest,hard-working ancestry (1893-1'960, noaelist) -lohn Philtips American Marquafld The ways in which numbers behave and interact reveal the most eisential workings of their archetype. A look at the actual numbers and prbcessesinvolved in pentagonal selfs1'mrnetry will take ui as close as we can get to understanding the piinciples of the Pentad. The key to this regenerative pentagonal giometry can be {ound in a particular. selfg"r,"ritit g ttirmbe. ieries known as the Fibonacci (fib-ohNAH-chee) sequence. Until the twelfth century, EwoPean commerce,banking, and measurement relied upon the clumsy system of Roman numerals, a vestige of the ancient Roman Ernpire, to calculate with. Try multiplying or even adding two Roman numerals together to seehow difficult it is even to begin. But Of in the year 1202a book, LiberAbaci(TheBook Computation), was published by a mathematician and merchant, Leonardo of Piia. This booi convinced Europe to convet to the numerals we use today, zerc through nine, known by their origin and Dath to the West as the Hindu-Arabic numerals. Because his father was nicknamed Bonaccio ("man of good cheer")' "hlius Leonardo was known by the Latin for sonof Bonaccio, Bonaccio,"contractedto "Fibonacci." Of interest to us from his book is an ancient number puzzle about breeding rabbits. Although there's no evidence that Fibonacci suspected it this Puzzle holds the key to selfreplicating growth in geometry and nature. The "rabbit riddle" essentially goes like this, The date is January 1 and a pair of newbom rabbits, a male and female, are in a pen. How many pairs of rabbits will there be one year later if newbom rabbits take exactly one month to mature, at which time they immediately mate, gestate for another month, then give birth to another pair like themselves, and mate every month thereafter? Every newborn pair repeats this pattem of rnonthly maturing, mating, gestating, and breeding other identical pairs, which likewise do the same. (We assume an ideal situation: no pair dies or deviates from the pattern.) A visual sketch of their family tueeis tJrebest way to sta*.



We soon become bogged down in the drawing of s1'rnbols representingrabbiipairs. But Fibonacciwanted us to notrce ilLenumbelsrepresenting their growth. A count of the newborn, mature and total rabbit pairs each month produces an interesting pattem, which recurs for each pair' \44ren we notice its secret-that each two consecutive numbers add together to produce the next number-we can solve the puizle easily by continuing this pattem to the next Tanuarv1. fftit same pattem would recur if someone decided to soread a rumor in a crowd' If he thinks about it for thirty seconds and then tells it to someone who hasn't heard it every thirty secondsthereafter, and if everyone who hears it waits thirty secondsand then also passesit along every thirty secondi, the numbers of knowers, tellers, and hearers will increase by terms of the Fibonacci sequence. Other pheRule 1 I;r one month each newmatu(e. born pair becomes Rule 2 Each mature pair breeds a newborn pair and mates again.

o O

Newbom pair Mature pair

Follow the two "rules" of growth to find how many pairs of rabbits there will be on January L. Notice, in the fashion of self-similarity, that if you break off arry "branch" it resemblesthe whole family tree.

January February March Anril

Total Pairs 1 0 1 0 1 1 1 . 1 . 2 t z J

May 'qrqqer ;o slrudgg7aqllllr araql 1 l,renwf u6 :raa,rsuy

2 3 5 8

3 5 8 13 1

5 8 3 21.

June Iuly August


nomena from handshakes in a crowded room to robotbuilding robots would do the same. The Fibonacci sequenceactually begins with two terms, zero and unity, nothing and everything, the Unknowable and the manifest Monad. Theseare the first two terms. Their sum, another unity, is the thfud term. To find eachnext term, just add the two latest terms together. This processproduces the endlessseries0, 1,,1,, 3, 5, 8, 13,21,, 55, 89,1,M,233, 2, 34, 377.610.. . . Look at this series.At first glance we seea chain of numbers. But look beyond the visible numbers to the self-accumulating process by which they grow. The series grows by accruing terms that come fuorr.within itself, from its immediate past, taking nothing from outside the sequencefor its growth. Each term may be traced back to its begirming as in the Monad, which itself arose from the incompre""ity hensiblemystery of zero. This principle of ongoing growth-from-within is the essenceof the Pentad's principle of regeneration and the pulsing rhyttr-ms of natural growth and dissolution. It appears in plants, music, seashells, spiral galaxies, the human body, and everything associatedwith the fiveness in naflfe.

Genealogy of the drone (male)honeybee.Some irsects and microscopic life expeience pafiheno gmesis ("virgin birth") whereby unfertilized females give birth to males. But fertilized females always give birth to other females. This pattem maintains the d1'namic balance of the hive. The family tree of any bee branches in the accumulative Fibonacci growth rhythm. And each branch resembles the whole family history.



(('('( The average number of petals on each type of flower in a field will be Fibonacci numbers, as are the numbers of pine needles accumulating as clusters in different Pine sPecies.

Sneezewort (Achillea . ptarmica) Eachbranc}:. lengthens through time and then reproduces another like itself, which repeats the Pattem of Fibonacci branching.




# Petals 2

3 5 The piano keyboard is a metaphor of accumulating vibration and so is structured by terms of the Fibonacci sequence.The thirteen-note chromatic musical octave consists of eight white keys (whole tones) and five black keys (sharps and flats) arranged in groups of threes and twos making one full octave.


55 89

Flower Enchanter's nightshade kis, lilies, trillium Al1 edible fruits, delphinium, larkspurs, buttercups, colurnbines, milkwort Other delphiniums, lesser celandine, some daisies, field senecio Globe flower, ragwort, "double" delphiniums, mayweed, corn marigoid, chamornile Heleniums, asters,chicory doronicum, some hawkbits, many wildflowers Common daisies, plantains, gaillardias, pyrethrums, hawkbits, hawkweeds Michaelmasdaisies Michaelmasdaisies




one treasures: is the Write out the first few terms of the Fibonacci sequence' ft".""aer eachto makeit a fraction,andundemeath Theorem PYthagoras; Jt"* the of " shiftedback oneterm' write the Fibonaccisequence of the other, diztision a line 55 34 and into extreme mean 34 2l ratio.The first we maY of to compare a measure Use a calculator to divide and convert each fraction into we gold;thesecond maY decimal form. closer to jeutel. namea precious A graph of the results shows each termgetting

t'uro Geometryhas great

6 + + ?t g #

. an iaei ot f.ef903398875 ' . or rounded off to 1'518or even ii begins crudely, pulsing far over then 1.62. Notice how

-Johannes KePler

1.63 1.65 t63 162 161 1.53 t.51 1.56 1.55 ),53 t52 l.5l 1.50

The Fibonacci sequencehas fascinating number ProPerties. For examPle, its Part resemblesthe whole. Use a calculator to divide the 1.6180339.. twelfth term, 89, into uniry The result is an endless decimal that, broken into parts, replicates the entire 1/89 = Fibonaccisequence: ... 0.011235955056179 = the sum of

Historical names of the Golden Mean: thesection Plato and theextreme ffieanntio EucIid aurm sectio(goldm section) Romans the diaine proportion Luca Pacioli

ChristopherClavius Godlikeproportion fohannes Kepler thediainesection

JohannF.Lorerrtz J.Leslie Adolf Zeising Mark Barr

dioision thecontinued section theffiedial thegoldm cut

.0 .01 .001 .0002 .00003 .000005 .0000008 .00000013 .000000021 .0000000034 .0000000005s .000000000089 .0000000000144 .00000000000233 .00000000000032 +...

o (phi)



under then over the ideal, getting closer and closer on-its wav toward an inJinite at which it will never arrive' We Lir" coa,theDiuine *uich us its decimals get longer and longer, like a thirsty is Proportion alwaYs taproot reachingfor the infinite that beckonsit onward' I he to similar itself. farther out you go in the Fibonacci sequencethe more precise the result becomes as it hones into the ideal' -Fra Luca Pacioli The ideal it approacheshas had manyrlames over the centuries,often eipressing highest regard for it with terms including "golden," "divine" and even "Codlike'" lt is the prouerbiil i'qolden mean," the ideal balance of life' hresentlyit isil'rnbolized by the Greekletter Phi (O),named in this century to honor the Greek sculptor Phidias, who used it to proportion his designs, ranging from the Parthenonto his famous statueof Zeus. . We Actually @is not a number but a rclationship tend to Eorn o, the focus on the visible numbers, but they rePresent t}:.eaccu' finite an encloses infinite series, mulatiaeproicusthat manifests them. We could actually start with any two numbers, not just zero and one, adding each And in theunlimited consec;ive pair to get the next term, yet the ideal limit will limitsappear, Try always approach <D. it. Start with any two numbers and Sothesoulof immensitY build youi own sequence using the accumuiativerule. Use consecutive pairs as fractions, anc' a calculator to divide the in dutells minuta you will seethe seriesalso approachthe ideal (D.Different And in thenaftowest becomemore exactat different rates. Iimits,nolimitsinhere. sequences we approach @it leads us to its source in the Any way joy the What to discern infinite. Heie are two equations mathematicians have for;nd in minute infinity! that also approach the ideal value of O. Don't be scaredor in Theaastto perceiae the anxious ab-oit them. Don't even try to solve theseequations.

small, whatDiainity! -Jakob Bemoulli (1654-1705, Swiss mathernatician)

o- 1 +l o I +


1 l + l

Mathematical formulas for iD are composed of an endless chain ofparts that resemble each other and the whole.



+ yt +\nrffi-



Just look at them. Each is a picture, a mathematical rnandala

ciothing the infinite asit zoomsby. Notice how eachis composed iolely of unity interacting wiih itself, embedded Monads unfurling as far aswe can seein a self-replicating rhythn, like mirrors facing each other whose reflecting image gets smaller and smaller- You don't have to see the whole to know it all. If you cut off a "branch" anywhere, the part resemblesthe whole equation! We can make this relationship visible by geometrically dividing a line at its golden mean (follow the steps described with the illustration). \A/hen we cut any line we think we have produced two, a Dyad. But Dyads representillusions. Whole Golden Mean Small part

o (Phi) Large part

be cafinot I wothings rightly put together must a without third;there of bond union besome And the them. between is fairestbond thntwhich comPlete the makes most fusionof itselfandthe it which combines; things is andproportion best a to adapled ffict such union. -Plato

The golden mean is the only way to divide a line so that the whole and part, simultaneously relate to eachother in the same way.

part Whole -- Larye Large part Smallpart


How to divide any line at its golden mean.

Start with any line and construct a square upon it. Quarter the square and draw a line from the comer to the midpoint of one side as shown, making a triangle. Put the point of your compass on the right end of the diagonal, and put the pencil on the sqriare's bottom comer. Swing the compass up until it passesthe diagonal (complete the circle). Then place the point of the compass on the other end of the diagonal and open the compass to the point just made. Swing the compass downward to seeit cross the original line at its golden mean balance.

122 (Din the pentagram. The pentagram star is itself a neatly packagedvisual expression of the golden mean, everywhere crossing itself into O relationships. Each part relates to all others and to the proporwhole in a chain of <D tions. I4/hen we look at any five-petaled flower we are seeing the flag of life, the selfreproducing force of life made visible as geometric harmony. Lr every flower we are liierally gazing into the face of the infinite.


rtftffi fttr# Every cut actually creates a three-it-one Borromean bond between a large part, a small part, and the whole that unites them. A division at the golden mean brings these three parts into special harmony, proportioning thern uniquely so that the lengthof the wholeline is to the largepart in the same rctio that the largepart is to the small part. This is the beauty of the golden mean. It maintains a single relationship among the parts with each other and to the whole. This is the mathematical definition of harmony and balance, the truth of which nature embodies and we admire asher forms. It is the core idea of nature's self-replicationand sits at the heart of the arche$pe of the Pentad. Buitd Gotden Meon Cotipers Enough with numbers, fractions, and formulas. We now know enough to construct a mechanical device that will Iocatethe golden mean for us automatica-Ily. Actually there are two devices; they are known as "calipers." They were usedby Renaissance artistsfor laying out the proportions of their compositions on canvas and in stone. We can use them for exploring (D relationships in geometry, nature, and art without resorting to numbers or calculation, Becausea golden mean caliper is constructed on the frame of a pentagram, O is built into it. Mathematicians call this type of device a "furkage," or a "pantograph." It is used for making enlargernentsor reductions of a drawing by O proportions.

Whrn or, ,rrc Eternity in thingsthatpass azuay, then haspure one knowledge. -Bhagavad Gita



You'Il need strips of firm material, like stiff cardboard or plastic straws, and metal brads, pins, or small bolts and nuts' If you have the skills and tools, finer models may be built of wooden or metal rods with bolts and butterfly nuts for easy loosening and ti ghtening. First, construct a very large pentagram star. Instead of an ordinary compass you may wish to use stdng attached to a tack and tied to a pencil. Lay the strips (or rods) along the diagonals of the star as shown. Mark and cut the end points. Poke a hole at crossing . points through which to insert the brads (or bolts). Notice that brads are placed ashinges at the vertex points and intersections of the star, so the leg must be slightly longer than the squared edges. Link the striPs as shown in the illustrations. The simpler golden mean caliper, made of two lengths joined at their golden mean, will indicate at its large side a distance @ (1.618.. .) times larger than the distance between the pointers of its smaller side. The second type of caliper will divide any straight line at its golden mean. Just put the extremes at the beginning and end of any line and its middle pointer will indicate the golden mean. proportions Both types of caliper will let you find the <D in the pentagram and decomposed pentagon.


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A @ l


Goldenmeancalipers always showa changing, relative<D relatiorshipbetween pointers.


will Goldm meancaliPers in revealthe O relationshiPs pentagonalgeometryand nature'sdesigns.

small Harmony mnkes things grow. Lackof it grcat things decaY. makes -Sallust (c. 86-34 e.c., Roman historian and politician)

Explore O relations in the pentaglam star.

Dlscover O in the Humon BodU

of Mon i, themmsure all things, of things that arc that they are,attd of thingsthat arenot that theyarenot. -Protagoras of Abdera (c.485-410B.c., Greek philosopher)

The ancient Greeks discovered, or more likely leamed from the Eg;rptians, that the human body is ideally structured in part and whole according to the golden mean. It's well known that if the measutements of many people are taken and averaged, the position of the navel, our first channel of nourishment and life, divides the body's entire height from crown to soles at the golden mean. Women's measurements seem to apProach the ideal more quickJy than men's. O is a proportion found in nafural growth. It increases geometrically simpiy by adding each two last lengths together to get the next. Or you can use the golden mean



@l[6 II


L i, t ighty airhonorable Soul for a Reasonable to Iiaein soDiainely built a Mansion as the Bodyshe resides altogethu in, unacquainted the with exquisite structureof it. -Robert Boyle

The navel divides the whole body as the brow divides the face.

calipers to extend a line into larger and larger <Dsections, This pattem is the frame of ideal expansion found in the whole and in great detail of the human body. Everyoneis a little stretched or compressed from the ideaf but tire statistical averagetends toward @. The basic unit is the vertical distance between the brow (the top of the eye) and the tip o{ the nose. The distance from the brow to the crown is @times larger than the brow-nose unit. The brow seemsto be a tuming point, a plane of reflection. The O ratio is also found as the distance from the nose to the base of the neck. It next extends from the neck to the arm,pit .there to the navel, to the reach of the fingertips, and finally from the fingertips to the soles.Thus, the 6ody is ideally divided into seven (D sections. This design was consciously used in ancient sculpture when depicting the ideal bodies of.the etemal gods. Human figures were "purposely depicted less perfectiy. _. F" O proportion was applied in the scuipture of Phidias in various width relationships: head/throat fore_ calflankle.

Mor,kno* thyself in trueproportion. --Oracle of Delphi

^ry:l*:t?t,widest part of thjgh/narrowest of thigh, Thewrist divides our cubit part at the goldenmean.



Usethe goldenmeancalipersto verify theseQ relationshipsin youi own body. It's possible,asI've done,to build a much larger set of goldenmeancaliPersthat indicatesall levelsof thebody at once. these Youcanseethis additive relationshipeasilyin the bones of your hand.Simply matchthe first two bonesof a finger of onehand with the third boneof the samefinger of the other

Ideal human proportions expressa golden ratio sequence.

T hr rorp rr rt ret rrt cheth out his rule;hemarketh it out roith a tine; hefitteth it with planes, he and tnarketh out with a it compass, makethit and afterthefigure of a man accordingto the beautyof aman. -Bible (Isaiah 44:13)

The hand begins io replicate ihe O progression of the whole body. In Arabic calligraphy, the shape of the hand spells "A11ah."

I,Vhenthe length of each finger joint is multiplied by 1.518.. . , the length of the next larger section is indicated.

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You'll need stiff cardboard (index-card material will work)/ a cutting tool, a needle and thread. Consfruct three identical golden rectangles. Cut slits aiong their centers as shown, long enough to slip another rectangle'ssmaller side through. The third rectangle, whose slit extends to one edge, slips around both to createthe three-dimensional frame. You miy wish to run tape aiong the ioints to strengthen the frame. Using a needle and thread you can sew through five corners to reveal the pentagram star (see iilustration) that sharesthe (Dratio with the golden rectangles. By cormecting with thread only the midpoints of short sides, you can outline the edges of an octahedron. And by connecting every comer, you can make the twenty hiangular facesof an icosahedron. Wiih ingenuity, you can also build the remaining Platonicvolumes on this frame. Discover O in Ctossic Art, Crofts, ond Architecture From their deep study of geometry and nature the ancient mathematicalphilosophersinfluenced artists, craftspeople, and architects. Since ancient times the (D proportion has

lines Regutating . . . are ...aspfingboardandnot . a straightjacket. . theY . satisfytheartist'ssense . . on nnd confer theworka qualityof rhythm. -19 -Le Corbusier (1887 65, Swiss architect, city planner, and writer)

law of I(nowledge a basic giaesafeeling of sureness the whichenables artist to put into rmlizationdreams would rahich otherwise in haoe been dissipated uncertainty. -Jay Hambidge

This Babylonlan stela depicting an initiate being led by priests into the presenceof the sun god contains many (D relationships.



This sym-rtry cannotbe used unconsciously althoughmany its of shapes approximated are by designers great natizte of ability whose sense form of is highly dezteloped. -Jay Hambidge (1867-L924, American architect, geometer, educator,and author)

Art without knowledge is nothing! -Jean Mignot (Gothic master builder from Paris, addressing architects in Milan, 1398)

This circular Greek libation pan achieves visual and geometric balance with its handle by the golden mean. Togethei they fit within the frame of a golden rectangledivided into a squire and smaller golden rectangle.

The west facade of the Greek Parthenon fiis within a golden rectangle. The United Nations Building in New York was designedas three golden rectangles, like three Parthenonsstackedupon each other, appropriate to the mission of promoting world harmony.



The west facade o{ the Cathe-

dral of Notre Dame in Paris is rich in golden mean relationships. Use the golden mean calipers to verifY them and find more.

shown up in artistic cornposition, it's simplest expression The most complex designsemploy being a golden rectangle. pentagon Somedesignsare higruy of the diecomposition the complex, such as those underlying Codric cathedrals ano They are comprehensiveand of Dain'tines the Renaissance. iutirfulig, evm uplifting, due to the harmony inherent in *," n6o*1trv of self-replication to which our eye responds' Lse the golden mean calipers to verify and explore the O proportiois in the following cornpositions' Discover Pentogonot Composition in Art Of alJpeoples,the ancientEglptians were the most brilliant at integrating geometry, symbolism, art, myth, ancl lar*us" fi theii works. For example,their hieroglyph for the luoi. A. "underworld" within ourselves,is a star within a circle. So whenevet they created art that related to the duat ttey composed it on a large pentagonal grid decomposed into smiller pentagons and pentagrams. There arg many examples of the O proportion in Eglptian art and architecture.

is architecturenot, S acred to asour timechoosessee it, a "free"art, deaeloPed and from "feelings" but " sentiment," it is an tied art strictlY bYand deoeloped thelawsof from geometry. -Fredrik MacodY Lr':nd

proportionl lT hegotde, of is a scale proPortions the makes bad which and dfficult [to produce] easy. thegood -Aibert Einsiein



The gold mask of the sarcophagus of Tut-AnkhAmon, which contained his muruny, was appropriately designed using the selfsimilar geometry of the pentagon, sl.rnbol of regeneration and rebirth.

The hieroglyph of the daat (Eglptian for "underworld") underlies the geometry of Egyptian art concerning journeys ihrough the underworld.

Pentagonal symmehy allows each small part to be a whole in itself while harmonizing with other parts and the greater whole. The visual effect is pleasing,balanced,even beautiful, and conveys the star's feelings of excellence, power, authority, life, and humarmess. Use the golden mean calipers to find the @relationships harmonizing the elementswithin eachcomposition. With the publication in 1509of Fra LucJ pacioli,s De Draina Proportiane,which rnarveled at the geometric relationships within the pentagory artists regained accessto the star's conskuction and to its value for organizing spaceaesthetically in harmonic composition. As tlorne,Jdirector of antiquities, Raphael (1,483-1520) was responsible for draw_ ing newly unearthed art, crafts,and archiiecture.He noticed


Salvador Dali was a student of ancient design ald used pentagonal syrnrneky in many of his paintings.

The good, course, of is always beautifuI, the and beautiful nez;er lacks proportion. -Plato

Jewelry of Tut-Ankh-Amon from Thebes.The rectangular pec_ toral symbolizes the creation of the universe by the sin above the -watersof chaos,while the nearly hiangular pectoral represents rne Duth ot the sun and moon.Both were designedusinq pentas_ onal symmetry. Note how the sun in one and ihe moon i"'tt"" " other are at the center of five-pointed stars, forming the glyDh of the d df, the "underworld,, path to rebirth and ."gjrr"rrEo.,.

Beauty dothofitself persuade eyes men the of without orctor. an -William Shakespeare



aside, But let usleaoe tion the Glaucon, propor lines. . . or we between our shallhaoe fill of many timesthenumberof discussions hadbefore. zae -Socrates (470-399 8.c., Greekphilosopher)

that the ancient Roman works emulated the Greek and Egyptian symrnetries. From this observation and his many teachers,Raphael learned to use the pentagon's compositional harmony which he, like many artists of the Renaissance,applied to quite a few of his paintings. His Dispute ooer the Holy Sacramentuses the regenerating pentagram inscribed in the circle. The subdividing pentagons and pentagrams provide a frame of regulating lines along which both the figures and architectureare aligned.

o Our final look at the principles of the Pentad draws us into the most widespread shape in nature-the spiral. Spirals are deeply rooted in the architecture of the universe; they are found in every size and substance. We're aiways intimate with spirals yet rarely notice them. Sometimes we miss them due to familiarity, as in water whirling down the tub's drain and in the shape of our ears. Sometimes we miss them because their obscurity,as in the spiof



fal "stafucase"of leaves whirling around a stem' Sometimes we miss them because of their size, or distance, hufflcanes or galaxies. And sometimes we miss them because of their invlsibility, as in the shape of the wind and waves of emotion. \iVhat makes spirals so prevalent in cosmic design? They are the purest expression of moving energy' Wherever energy is left to -o:rre on its own it resolves into spirals' The univ"eise moves and transforms in spirals, never straight lines. Spirals show up as the paths of moving atoms and atmospireres,in molecules and rninerals, in the forms ol flowing water, and in the bodies of plants, animals, humans, and the greater bodies of outer space.A universal integrity of soirals r.rnitesall creation' the Like the pentagramstar,the spiral expresses geomeapgeal to b.e try of self-similarity.At fust the spiral doesn't pentagonal.but wherever you seea star you will find a spiwe ial rotLd within it. \ y'hen feel its hlpnotic lure while gazresponding to the Pentad's siren call ing into a spiral we are into the self-sirnilar infinite. The spiral's role in nature is transformation. Similarly, in myth and religion it is the path of spiritual and mystical transformation. you You may be surprised at all the spiral processes at plants, already know abou!.now you may leam to look fruit, vegetables, and other familiar natural forms differently. ihree principles of nature's spiral will reveal to us its secretsof universal construction. @ Spirals grow by self-accumulation. (6; Every spiral has a "calm eYe." resolve into spiral balance. @ Clashing opposites

SPIRALS UNRAVELINO bg Growth Setf-occumulotion Although mathematiciansdistinguish many speciesof spirals, they all have in common the fact that they wrap around a fixed point at a changing distance.



It is perhaps more a a fortunate destiny to hazte taste collecting shells for than to beborna millionaire. -Robert Louis Stevenson (1850-1894, Scottishessayist, novelist, and poet)


We can begin to understand spirals by doodling them. Draw some spirals with your normal writing hand. Then draw spirals with the opposite hand. Then read on. Did your spirals spin in the same or opposite directions? Did they flow outward or inward? Did the distance between coils stay the same,widery or narrow? Adding to the definition of these shapes,we can say that different spirais grow by curling around their points at different rates. The two main types of spirals we're most often exposed to are called the Archimedian spiral and the golden spiral. Named after the Greek mathematician and scientist Archimedes,who describedit in his book On Spirals(c. 225 B.c.),an Archimedian spiral is one whose distancefrom the point grows at a fixed rate.The distancebetweensuccessive coils is always the same.We encounter the Archimedian spiral as a coil of rope, clock springs,record grooves,and a ro11 of paper towels. The helix is a three-dimensionalversion of the Archimedian spiral, found in bolts and coil-springs, in candy-cane stripes and barber poles, and in the double helix of the DNA molecule, our genetic heritage. But the spiral most commonly found in nature's public manuscript is another f1pe, the golden spiral. Unlike the Archimedian spiral, the distance between the golden spiral's coils keeps increasing, growing wider as it moves away from the source or natlower as it moves toward it. Called by many n:unes through history, it was apparently weil known in ancient times as evidenced in prehistoric art and worldwide ornament. This is nature's spiral of seashellsand ram's horns, of our ears and fists, oi waterv whirlpools and star-spangled galaxies. It grows from within itself and increases according to the Fibonacci processof accumulation. A simple experiment will reveal the important difference in the ways these two kinds of spirals growFotd the Archimedion ond Gotden Spirots Cut two ships of papeL On one, mark off distancesat one inch (or one centimeter, or one finger width, or any unit you may choose),at two inches, tfuee inches, four inches, and so



1 1 2





ilfr-=illl // tl llllu-/ll ArchirnediansPiral

tr=1\ ll a) tl

u-=_= Golden spiral

Nautilus shell

Red cabbage


on. In other words, make the counting sequencevisible. Fold a right angle at each mark so that the striP tums around to become an Archimedian spiral. The distance between its coils remains constant, always that of a single unit. Its accumulation comes from outside itself, always adding an external "one" to the previous term. On the second strip, mark off units measured by the \l/henyou .. sequence1,L,2,3,5,8,13,21,. inches. Fibonacci and fold the strip you'll seethat the distance between crease This is the golden spiral found its coils continually increaseS. in nature. Like the {amous Fibonacci nunber series, the golden spiral grows from within itselt';nothing is absorbed from outside. It is the physical representation of self-accumulation. Its appeatancetells us that the Fibonacci sequence

Clock spring

Paper towels



and @ undertie an internal dynamic balance at work.




o The Swiss mathematician Jakob Bernoulli (1654-1705), Datriarch of a family of distinguished mathematicjansand icientists, devoted a great deil of study to this particular spiral. He discovered its self-accumulating,self-reproducine nature and gave the spiraI a motto (perhapsthe only one mutatoresurSoasiociatedwitliu geo*ut i. shape):Eadem Although changed, I arise again the same' He. ryas .so impresied with the properties of the golden spiral that he reouestedthat the shape and its motto be carved onto his tomb. Un-fortunately, the stonemason mistakenly carved an Archimedian spiral. Perhaps he didn't realize the important difference between them, or perhaps he just didn't know how to geometrically construct the spiral Bemoulli had Here's how it's done: reouested,

6\ )


unfurt NotureS Gotden Spirot from o Gotden Rectongte are in a <D The solden rectangle, whose long and short sides 'bibonaccr relati"onship(or approximated by Iarge consecutive Atter connumbers), holds t"rat[e'" spiral coiled within it' point ofyour rt u"tl"e u eofa"n rectangleof any size,put the it to the length of tlre snort compusi oi a comer and open tne side. Swing the compassto measurethis distanceonto it to mark and swing ionger sidi Turn thJcompass aror'rnd, onto the other long side' Corurect the o? ?t tu." distance ooirtt," to -uk" u square within the golden rectangle Astonishingly, what remaittsis a smaller golden rectangletumed on its side. If this processof marking out a square trom eacn new solden reclangle'sshort side is repeated,a smaller' tumed iolden rectangle will always remain' Lop out a square iuitt itt it utta uttothergolden rectangleappears This Process of squares chasing i single receding golden mathematically endless.Theoretically, there is no.end to ttre trail of squarei cut out of a continually resurrecting goiden rectangli seducing the squares forever onward' Corurect corres;onding points on each successivewhirling square ih"r .;tirs, intemal comers.,or extemal comerg and "i."esliral appears as a smooth curve. Life's proportion lures the


a gotden rectangle shows now nafural forms like whirlpools,' dissolve,, by removing self-sirnilarpirts. Adding squaresonto the targe outside of a golden rec_ rangleshows how growth occurs through the-accumula_ hon of self-similar parts.




matter, symbolized by the square, l"ft b"hi";-;'i;;l;." Discover the Spirol in the Stor in ptonts

Jrs#. ::t F:;;ffi,?ffiiil:',lX:: :.#l':hiiii

rectangle, another goldenr""a-*ru.,"r,??.lroduced Solden '"'r'tnurl towardthelargerinfirite in perfecio p.or;;;;i:. ,Th9spiril risults from the play of the rationalsquare with the transcendental r O ctothed the four stares ll?".ll,lt.h" play of life iiself, by or

adding ano{het square ,h";;ir-il""oi ;" ont6 i^.-11h":Jly targersideof eachsuccessiv

larger,the spiral exhibitsexpansiue ." ^ 9lo**g growth in -"*p*l of seashellsand hurricanes. Wu :T,I"-ri rectangle u goroen by addinsa square "ur, ontoits lafi;;fi ih" resuttis another larqerself_similar but golden,l.t*!i"..ny

therriu"rr", ua,ui" dissolae o" o.,, ;;ilff;* ffi ilfftXf*i:.oo* "r""t ,tLlyi^y:

tle square ahead along a spiral. This diminishing path h.y.":enrs.grJw

smalle and r

#;,i,,3-':t:1.,:"10-,. "r"o,"rr-,Jli?,i:: aroglj one.another thesarire as s;l;";;;;;i-"f *,#;:"" tr"ry hi*gte in apentagranii. u goijunt i*gf_]mrr, _.,_ we when see five_petiled a fl"c finda spirarassociated it, as*" o#:ril;:XilYivs with Youcansee spiral*a the .i_,irtrrlorlfu"'iu ro.rr._ "i",

triansre,, sorden_

of self_accumutafion -frl*iiJ, and dissolu_ *: are not limited non :lt:lhg,.Orocesses rectangles to *a ,qu"."r. whose larger and shorter sj




The self-similar logarithmic spiral is found in the selfs'irnilar Pars of a golden triangle.

A "sun's-eYe view" along the stem of a weed or anY Plant or tree reveals its whirling pentagonal structure.

of This sliceof a celerybunch showsa spiral arrangement stalks star. as accumulating a Pentagram

ine at plants from the "sun's-eyeview" shaight into its stem Common weedsare excellentfor this' Squint a litor"branch. tle and you'll seethe irregular overall fiveness-of the-Ieaves and branches as they wind up the stem. If they form"q.u P"lfect pentagram the leaves would overlap' But a slightly irre$rlar iar shape permits each leaf to be exposed to the sun. lettuce,or any plant in PeeIback the leavesof a cabbage, the reverse of the sequencein which they Srew and you'll seethat they form the points of a stilted Pentagram star'

Spiraling bracts of a pinecone.




w 'l.lX jt tt

INTO TTIE"EYE" OF THE STORM Ege" tlove o "Cotm Spirots Noturot and A curiousfact about the golden spiral in mathematics a charactercllrnature is that it hasa "calm eye," a corewith A watery ?".""r fto- that of the whirlwind around it' *rups uror^d a core of air; the leavesof a -tree ;j;l;.;l , *oody trunk rhe calm eye correspondsto ;'h;;i;;;J tii "."ro" at the verybeginning of the Fibonacci.sequence' Like the deceptively calm eye within a storm or hurrrcane/ ioi"nr" u.tiori t*irls all around it but the eyeitself,remahs rustlein the wind' but the trunk Leaves olacid,untouched. is rooted in the earth. ofthe universeThe Ftindu religion refersto tr'voaspects ani the manifest The unmanifestis called the unmanifest unknowable "Brahmawithout qualities," Nirsuna Brahma, while the manifest universeis cal1ed *iif,."i js "tt"t""t"ristics, SasunaBrahma,"Brahmawith qualities,"all that knowabie. These correspond to the eye and the spiral' the unknowablesourceand the turbulent universearound tt/ d mvsteriouslv rivenby the "calm" core' ' an-asymptol.e' cail Mathematicians thegoldenspinl's,eye but never reached Althougn to a olace alwavs approached "whiriittg itt the golden rectangle.the eye is th'e squares alwavi inJinitely ahead,we can pinpoint it exactly' Io locate the eye in u goidett rectanglejust draw one diagonal from corner to comer within a large golden rectangle u"9 q:" another diagonal in the next smaller golden rectangle' lhe eye is locat"edat ihe Point where the diagonals of every golden rectanglecross. " Leonardo?a Vinci wrote, "A vortex, unlike a wheel'


I \



Whirlpool fr.rnnel


Fingerprint "plain whorl" type spiral

The planetsroll on the curved surfaceof a whirlpool called the soiar system with the sun at its eye'



his hath Eaery storm calm. -Robert Greene (1560-1592' English poet and PlaY'wright) \


eye Findingthe spiral's in a goldenrectangle'

moves faster toward its center."He was referring to a fascinatine differencebetween thesetwo forms' Watch a bicycle slowlynearthe s wheeitum, you can seethe spoke movemore hub than at ihe rim: A spiral does the opposite, movtngfaster at its eve than it does further out. The solar system is a great whirlpool with planets rolling on its surface' Those closest to the' sun orbit faster (Mercury in eighty-eight days) than thoseat the periphery (Pluto in 248 years).In linlture w9 seethis charicteiistic in water whirling down the drain and h low-pressure storms twisting the atmospheresof many planets. lf you watch a daisy flower grow over time you ll iee its central vellow florets ("little flowers") emergemysteriously from ti't" tiuittg eye of the plant and grow, whirling outwird more slowly toward the periphery. In fact, the name daisy derives from "day's eye," the eye of the day. Leam to recognize spirals in nature, in the living t'trees" and in house plants' Look for the whirlpools we call "eye.'l Whatever its substance,the eye is always mystedous and hypnotically fascinating to watch and to contemplate as the source of the spiral. In watching water whirl down a drain, note that the narrower and calmer the eye is, the greater the sPeed and turbulence around it. Notice also that un]ike the center of a wheel, the spiral's eye is not fixed in one place but is dlmamic and flexible. Far from insignifican! the calm eye is the spiral's center of gavity around which it all balances. Without the eye

Jupiter's"eye" is a sPiraling storm many centuriesold.

no or there would be no spiral expansion dissolution, no whirling, no balance, life' A sphalis balance-in-motion without chanse tt lotcesdisplaving ;ril;"iibL;;gttpo



Underwater view of a whirlpool's fl owing circuit.

The digestive system of a Pig, like that of other animals, foliows a whirlpool to trarsform food into energy and waste.

demonchange, playing out before our eyes' Two activities shate this: g-du*"-it -tt otion: DroP thread or a toothpick- near the p"tiph"tv of u whirlpool of water going down a drain and same to notiie how asit travels around It continues pont In tne doesn't iirection. Its speed changes, but its orientation eye it will point upward)' irfin.r*tt *tt"i't it goesdolwn the This exiression of lts mathematicalself-similarity exPlarns planet whirling around the sun keeps-its axis *f,v " in thu ,urn" diriction tfuoughout its orbit' The the ""firtftt* off barth'sixis always points 23.5degrees the "plane" of solar svstem,while Uranus rolls on its side at 89 degrees'its oof"t rloi"ti"s toward the sun fwice yearly' This is the kind of "cha.tgewithout change" that inspired JakobBemoulh's epitaph. ' Self-replicating balance: Another way to seethis princi ole of balance tfuo=ughchange is to bend a piece of wire into 'a eolden spiral and"balancelt by a pencil or straw through th"e eve.Observehow it settlesinto a stateof balancewith its end pointing upward. Theru with a wire-cutting too,l tPP u leneih off tfie end and observe the spiral shift to rebalance itseif. Remarkably, the "new" spiral settles to resemble a miniature model of the original spiral's orientation, mahtaining change without change. We can aiso understand the spiral's balance by viewing it in sections. Although different in size, every segment has the sameturvature. If a microphotograph into the eye were enlarged, it would find another part of- the- spiral. upon whici it would fit exactly. No other spiral will do this, nor

Changewithout change. objectswirling arounda whirlpool An remains pointed in the same direction.



will any other spiral accommodate the dynamic balance that nature values. An Archimedian spiral's curvature gets broader as it widens, no section being identical with any other. are These characteristics the open secretof balance of plants, and galaxies.For example, animal horns, seashells, as the chambered nautilus creature grows larger, the gland that exudesshell material also grows, building a widening shell. The shell's golden spiral shape maintains the same center of gravity in any size, and so the nautilus need not relearnhow to balanceitself as it matures.The sameis true for the growing horns of a ram. As the hom material accumulates,growing larger and more massive,its golden spiral maintains the same center of gravity. Thus the ram need not adjust its posture throughout life to uphold the growing horns. Sirnilarty, the tueethat Puts out branches and leaves in spiral "staircases"around their respective"eyes" can get enormously large; yet the tree always balancesno matter how massiveit grows to be. In a whelk shell, the spiral's eye is the central column that gives the whole shell outstanding strength. Gothic and Renaissance artists used local shells to design spiral staircasesof great strength. The well-knortm sound of the ocean "heard" in a spiral seashellis actually the magnified echo of the blood pulsing in the listener's own ear. Divers observethat schoolsof fish, like groups of birds and swarns of insects,regroup in the pattern of a whirlpool spiral. Some say that this is becausethey are following the



unseenwhirling water or air. Others maintain that eachindividual fish, bird, and insect is like one cell, one part of a sinqle organism and they are linked as one energy field' " lryie."r,er the spiial appearsin plants, animals,or solar systems, it insures dlinamic balance during inevitable srowth and dissolution. Learn to identify and then contemilate the eye in natural spiral phenomena. Another property uniqul to this type of spiral is reflected in an altemate name for it, the "equiangular" spiral, coined by the French mathematician and philosopher in Ren6Descartes 1638:any line drawn from the pole to the curve cuts it at the exactsameor "equal" angle all around' The flight of a fly illustrates this. A fly's eye is madeof thousands of tiny, hexagonally packed facets. It sees the same tiny image in each facet, instead of one large image as we do. As the fly approaches an object, the image shifts slightly in each facet. In response, the fly shifts its body so tfrit tlie image remains fixed in its odginal place in each facet. As the distance to the obiectbecomesshorter, the angle of approach rernains constant. Thus, the path produced is The leavesof a plant form a continuous spiral around its stem (eye), the balanced structure of a whirlpool.

A fly approaches an object along a spiral path due to the structue of its eyes and nervous system.


151 Fish, tofioise, rabbit and other vertebrate embryos begin life as unfurling sPirals'


Acupuncfure makes use of the wav the spiral human embryo resembles the sPiral of the ear.

only different in made of segmentswhose arcs are idenl golden size-tracing the accumulative bones) we sPin Like other vertebrates(creatures into life in the womb as a spiral embryo We are conscious as other natenergy systemsthat whirl into con r:nfurling into life of uralinergies do along spiral lines along an endless path of sPiral tuansf s most obviThe outer ear, or auricle, is one of cowlicks, the along with curls of hair and ous spirals, mushroomlike vortex of our nostrils, o curling fists and accumulating fingemails' Chinese ing the principle that the part models tl whole, map the enthe spiral embryo onto the spiral ear. Thus, to treat any part malady of the body, they'll first treat the of the ear.

A*onis thewhole The offacts. encyclopedia of creation a thousand and acorn, is forests in one Rome, Egypt,Greece, lie Affierica, Britain, Gaul, in already thefirst folded man. -Ralph WaldoEmerson

152 The outer ear and inner ear (codrlea) are spirals. The shape of the cochlea, the organ that "hears" music, corresponds to how chromatic musical octaves appear when gnphed as wavelengths. Each note is idmtical to those directly above and below it on the spiral but with a oneoctave difference.


hns The manu)ho music in hissoulwill bein looe with theloaeliest. -Plato

Only in this century, thrgugh the pioneering work of Georg von Bekesy (1899-1972),have we begun to understand how we hear by means of the inner ear, our spirai (Greek for "snail shell"). The cochlea of each ear is cochlea encased within a hard shell (so we're not overpowered by our own voice) and filled with liquid. Around the central column is a "handrail" of sensitive hairs that decreasesin thickness as it winds upward. Sound vibrations enter the whirlpool of the outer ear and tap at the cochlea'sbase,traveling in waves up and around the inner liquid. Each tone makes waves that break at a different point up the handrail according to the tone's wavelength. The resulting spiral turbulence of the breaking wave triggers the sensitive hails at that location. The hairs then transduce the mechanical waves into unique elecfrical signals sent to the brain for intemretation. Humans hear approximhtely ten octaves of sound in a codrlea of two-and-three-quarter spiral tums. The coclrleae of other mammals have various numbers of tums and extended opporturLities for finer waves to break, which explains why dogs and mice can hear frequencies higher than humans can perceive. ' The first recorded fact of mathematical physics, athibuted to Pythagoras, is that musical notes canbe expressedas



rat ale sPiralsthat of The cochlea a cow,guineapig, mouse'and registerdifferent levelsof frequency'

eight)' Sowhen mathematical ratios (seechapters seven and ,he chromatic musical octave are graphed ;il;;;a accumulating accorains to iheir frequency of vibration' as shapeof the golden spiu't" gruph forms the ;;;;i#,h", the up irito threedimensionsit becomes snaPe ral. Stretlhed the organ that transduces sound to con"itn"...f'f*, the cochtei and all nature's spirals can thus be r"f""*"tt. or seenas solid, Iiquid, gaseous,and fiery rePresentatlons '.* the music that structures the cosmos' break exactlv'how do curling waves rise from the sea and belongs to the lore of ih" shore? This 6it of *itdo* ;;; the s"rfi.g tt o*f"age and what is referred to as "shooting "orskimmj-ig through the wave's eye' Windsblowing curl," its onto the oceansca"use liuge, rolling waves' The sharp eye of thu srrrfe, notes the distince betwien successive rows of


the thnto'er

ocean interminable wreathe ois?ed Your smiles. -AeschYlus


wheretheyagl h9 in ;""es break thecoctilea

. Behold theSea . . [in its] ebb flow, mathematic and gioinghintof thatwhich not. changes -Ralph Waldo Emerson



narrowlng incominq waves, their "wavelength"l Mtrhen-the hau the .waae-, depth to"iheboftom below the wave is exact,ly ano lenqlli between the waves, it will rise, Push torward'

its itself1nt0 eye it causing to curlarcund air mel"t resistance,

Spiral heart muscle

as an upright whirlpool io form a breaking wave of iine white b;bb"les. Why does it rise and Crxl at exactly hnlf the wavelength? For this reason you'll have to wait untii chapter eight"and the discussion of the "doubling" principle of musical octaves. Not only doesour whole body whirl intolife as a spiral, but at its heart,our body's literal heart, is a dexterous(leftspirmins) spiral vortex. Sandwiched in the middle of its tf,r"" tui"tt i" a sheet of contractor muscle in the shapeof nature's familiar golden spiral. Like the sun at the center g1gsiz'e of our personal solar system, this living whiripool, of youi spiral fist, contracts and relaxes with great strmgth usih" futt iliut "lub-dupp" beat sendingblood rushlS {r9m its left ventricle chamber (a smaller spiral whirlpool within The inner balance miles of blood vessels. it) through 60,000 we seek,-therefore, is already represented within the very fibersof our heart.

o For over a century scientists have known that the properties of the ninety-two natural atoms, and nearly a scoreof artificial ones,repeat in rycies or periods, like the notes-ofthe -Chemisls have arranged this cyclic list of musical octave. atoms into the well-known Periodic Table of Elements, usuallv represented as a flat chart in rows and columns showing aciumulations of electrons and protons, with their atomic mmbers from one to ninety-two and beyond But the "accumulation" of electrons is continuous, so the Periodic Table would be more accurately depicted as a continuous spiral ribbon, a widening vortex rolling around itself like a seashell,coctilea, galaxy, and the musical scale. The eye,or centerof gravity, of this spiral map_ofmatter falls along the colu{nn of inert gases,sometimes called nobie gases,becauseof their disdain for joining other elements in chemical bonding. They are the stable elements of the universe/ balanced and complete unto themselves. The farther

The Periodic Table of Elements as a continuous sPiral.



Svveou He Ne KI Xe Rn

Nave Helium (sun) Neon (nezu) Argon (inert) Krypton(hidden) Xenon (sftanger) Radon (ray)

AToMtc Nuvegn 2 10 18 36 54 86

Cuosesr NuNeen FreoNrcct 2 8 21 34 55 89

inert gas' the an atom falls on the Periodic Table from an by entering into. a it is to seek cornpletion 'ihe ;;;;";lil; closer the ele*l*t othei atoms' ,"iEn "tr'tipgas, the less needy they are' "tr"*t are to an ineri ments "i . gasesto A look at their ato-micnumbers shows the inert to numbers from.the have quantities of electronsvery close lf Fibonicci sequence. a spiraling strip of paper-representmg whirlpool were marked oft at Flbonaccl the Periodic Table loca["Xf*, al't"markings would fall on the approximate calm core' tioris of the inert elements,its AII the matter of the universe,representedby the Periodic Table of Elements, resemblesthe spiral structure ot smaller whirlpools that comprise this whole' Even the tinisubuto-iJ-particles" arJvortices that reveal their spiral "st sDin in bubble chambers. From the smallest particle eachd ancesthe whirlpools to the largestclustersof galaxies, same spiral step wien coming into, or dissolving out of' ,nanifestation.

A single electron in a suPerheated bubble chamber leaves spiral trails as it loses energy within the chamber's magneticfield.

THE DANCE OF SPIRALS into CtoshThegf,lesotve Spirots WhenOpposites What brings spirals about? In what conditions do they arise? Spirals arJ a iign of growth and trans{ormation through r6sistance. VVhiie their sizes and substances vary widely, nature's spirals result from an interplay of opposites, the clash of *ie Dyad. In its geomehic construction, we saw the spiral arise at ihe interface, where rational squaresendlessly cirasetheir opposites in the transcendental golden rectangle'

yawns atom Atom frotm as far or As Earth f'romMoon, stalfromstar. -Rahh WaldoEmerson



Thr rurrrrt thatzoith gentlemurffiur glides, Thouknow'st,being doth d, stopp' imPatientlY rage' -William ShakesPeare

stillness' The rnterfacewhere hot meetscold' motionmeets tyolg the-into rising crosses path of fallirg, any.of.thedua' I1'] spiral balance'bven love a dynamic resolve in "Iash"but ig".r-." und wisdom f ird balance the spin"o"a, ""J danceof life. ral "t the whirlpool F;"; h,'pesof spiral appear in nature: the vortex ir't" #it", the mushroom vortex rings' and variation on tne runoa"aau. e.trully, eachof theseis a ,truei.


" " eddv"(in o1dNorseidhasigrutied that

which flows 6ack around")' The MushroomVortex into sPjThis orinciple of oppositesresistingand resolving vortex' or rals is mosi easily understood as the mushroom them when you-pour:.movm.cliq51 ;;;"; ilgt. You'see like milk Lto a motionless liquid like coffee, or l"u 11" at poured liquid meets resistance its head and curls to tne tr"d".-il"fr ulso meets resistance and curls- further toanclcurllllg inward, repeatingthe cycleof meetingresistance disappears'leavmg resistance until ihe force diisipates and rs o",futf't"toitul puth.'Theresult appea in the form of amushshould be no surprise,sincea living m:tht:o.P room, which wrue is mostly water, too, although it moves-moteslowly displavine the samepattem as clashingliquids do ',q ieuJrsal of muihroom 'rortex rings, called "backwater by vortices," forms behind a flat plate aswater or air ru-shes it. Thesevortices are studied in wind-tunnel tests tor cars' boats, racing bicycles,and airplanes,where the spirais are "-turbul"na"." lf you've ever drjven behind t o*. ut

A iet of milk squirted into mushroom water resembles growth.




the another vehicle on a dusty road you may have noticed dual mirror spirals trailing the moving car as.rt trrusts throueh the still air. Dust makes the spirals visible' WE can see that the mirror spirals of the bivalve clamsheil,as well as the creaturewithin, and leavesdisplay vortex dngs. Only this dual golden spiral will -"rtttoo* allow each half to acciunulate material on its lip and grow Iarger in mirror fashion without inter{ering with the other half's increase.

{o Bivalve shell grows as mirror spirals. Many leaves unfurl as mushtoom vortices, which the ancients saw as the nostrils of Pan, the spirit of the living forest.



d e e9 Fratemal and identical twins as mirror spirals. The capital of a Greek Ionic column symbolizes the Earth s subtle energy flowing ttuough the temple, meeting resistanceat its head" and rolling back as musfuoom vortex nngs.

The mushroom vortex also describes the entry of twins in the womb, who whirl into life together. Since each must not interfere with the other's growth, they develop as vortex rings, $owing simultaneously larger while each maintains iti unique cmter of gravify. Fratemal twins arise from two fertilized eggs, while identical twins are bom of the same egg which has split. The spirals of fratemal and identical twins mirror.each other differently. The Vortex Street Betterto hearkento a brook, than watcn U*Ttt " i*r1"O"*rO

If you've ever been in a rowboat you may have noticed whirlpools hailing behind the moving oar asit cuts through still water. Look closely and you'Il see that the eddies are linked in a daisy chain of altemately opposite spinning whorls known collectively as a "vortex sheet." The Phbnomenon also occurs behind a still branch dipping into a moving stream.



in These phenomena

we waterandair maY, as think, betegarded the of letters a scriPtwhichit to is necessarY uselikethe of alphabet nature.Those whowish to remainat the of stage Pure gY phenomenolo r elinquish theabilitYto readthis writing and thus the its ability to understand see TheY the meaning. but Ietters, not wordsor sentences -Theodor Schwenk


ot' a interruPts movingstream, t9YTg, \ {hena still object formsto reestabbsn street distwbsstill water,a vortex object balance. The central axis of a vortex sfteet is a forward-flowing zigzag from which the ieparate vortices emerge and bal"tt" central rhythm of alternating Pulsation gives the "ti"". whole "street" stability. Each spiral whirls independently balanced,and ail spin in directions that suPport the d'rrection o{ the central sirearn. To understand its rocking rhythm, ihink of how you break into a run, ride a bicycle, or rollerskate: you start off rocking side to side to achieve balance while ihrusting forward. The more stable you become, the -and straighter the central axis becomes'This lesspoiarized is how the archetlpe struggies with resistanceto resolve itself into a state of dlnamic balance. The moving wings of a jet plane spiit the still air and leave a vorter street of sPiral turbulence behind them' Designers know that turbulence is a cause of drag on the vehicle, and they strive to reduce it. Fiocks of flying birds take advantage of vortex streets-in their farniliar V forrnations. Only the lead bird must really work at flapping its wingsi the others latch onto the undulating spiral wake of turbulence trailing behind it. They simply ritix their wings and let the rolling waves move them ui and down and forward. When the lead bird becomes

Birds rely on vortex streetsto make long-distancefl ights easier



uy ttte wind is a measureof its rurbuof ,6*" birds like owls are so perfectly .ies"ilrreJ-,h" l""""'il;;G, and ihey produce virtually no tubulence' surprise them to silently ;;t;;*1""";-r'otioa, ""uriittg tn"tlill1tn,^*.n"t ffee splits thebreeze,from your ringerto a vortex streetto spreao o. u b.l'ildit i, *ill iause an invisible behind it. ""'-oi" (deflecio tt tpin of the Earth and its Coriolis effect " object), low-pressure.storms l?"h :,f ll: ti." .f;;;;itig meshedwith high Pressure spin cor]nterclockwise, eouator the samerea.,ji*tt tfjit -"uther) spinning clockwise' For southof the equator'-losee.tne are.reversed stn, directions th9 laltn s Coriolis effect in miniattrre, that is, to see how a marble from ,f"t" atrnosphere to spiral, shoot rol^ *"t"t (as a spinning record tumtable toward lts center the rim of as a the South Fole)' The marble's path will cutl of ii;;;" in shifting . . . A.nd the qolden spiral like the great winds' thewinds,andin the ""- d;;;;; ;*eathei map and you'll see that high- and into cloudsthat arePressed low-pressurespiralsarelinked asa daisy-chainvortex streef and in bands aror.rrd the planet. These "iet streams"'rughheaaen betwixt seraice great vorearth,aresignsto PeoPle altltude zigzagging winds, are the central lanes of g between w hirling atmosPheric sPi i"* t ttu" tJwhocanunderstand' ""u""de"tin all meshed as such sPtrals' rne rals. Global weather is -Koran Earth's atmosphere is one whole' Changes in weather Pattems an)ryvheieaffect weather everl'rvhere' """Th;t;;;;aa"

rnoves ahead to wor:k at tired she falls back while another splitting the breeze for the others'

Sand dunes are carved by one side of the moving vortex street js we simplv call "the wind." Simitarly ribbed sand at the shore .u*"d tu the vortex streetformed by the ebbing water clashing with the"incomingwave.



The jet stream is the central zigzag of a vortex street ga(landing the planet. The sPirals that sPin at its sides are the meshed high- and lowpressure systems of our weather.

o7 gossip . .thebabbling theair.





Spiral storm

Differences in temperature, motiory direction, and moisture are resolved asvortex sfteets. Hot moisture rising above a cup of tea or hot soup meets resistancein the cooler, calmer air and curls around but continues to rise, forming a series of growing eddies meshing along a vortex street. A candle's flame wags back and forth, as a flapping flag does in the breeze, and for the same reasons,Both flame and flag move along the central zigzag current of the turbulence. Fly a kite with a long tail and observe the tail waggle as a vortex street, revealing the nolmally invisible rhythms of air.

Highpressure Lowpressure (storm) (fair weather) (north of the equator)




A tomado (in SPanishforsimifies "thunderonndo storm"fresults from a clashing of contrasting temPeratires and of winds. It is the fastestwind on earth approaching 300 mPh, caPable of destroying the instrummts designed to measure it'

PLANTS AS LIVINOTURBULENCE Believeone who knows: you will find more in woods than in will and books.Trees stones teachyou that which you cannever leam from masters. (1090?-1153) -St' Bernard Clairttaux of The struchlresof growing plants often resemblepattems of water and air in m'otion.And why shouldn't they?Plantsare mostly water, although they move much more slowly' so lt is appropriatethat they expressthe natural forms of flowing *uidt ett i-p"rtant Part bf our own processof exp.anding vision is to sel plants iifferently, not asstatic"things" but as dr,namic proctisrt, and to develop the ability to recognize oiants, in whole and part, as living whirlpools, waves' itrrshiootl rings, and vortex streets We can learn to read nature's "hydr6g1yPhics." lfany ptantsl ior e*umple, obviously resemble upright vortex ;tr;ets. The centril stem corresPonds with the zigzaggtng path flowing along the center of t: vort:x street.Each leaf, like a breaking wave/ emergesattemately from the stem and curls around, its farthest tiP tuming back in search of the elusive eye. The architectme of a plant resemblesthe Path of hot moisture rising into cooler, calmer, drier air. Its strucfure appearsas a vortex sheet,giving us a glimpse of the invisible pattern of flowing energy, the lines




Wffi#$F& \Artreat "Raceme"



The architecture of Plants and their parts exhibits the rock-. ing rhythm of a vortex street' rn also aPPears This cadence univalve shell as areas a cut that alternate along the central column, resembling leaves growing along a vine'

lW\ Com

An Grapevine Cut univalve shell


ru $t Peas



a of livins force along which the plant's cells manifest like bubblin"gfroth to dJfine its shape'Spiralsoccur in resPonse to the world's resistance. Plants and life in generalare a kind of living turbulence' Living vortices, like iiquid ones,have parts that model the wholJ. h largerplants and treeswe canseehow eachbrancn is a miniature vortex street emerging from the central trunk' Eachbough, branch, twig, stem, and leaf is a comPletvortex street ind part of greater vortex streets' \Arhenyou're out among plants in a field, park, or forest, practice seeing vegetatioX'as green fountains spraying up-!eav99 in a. slow whirl, like J slowly revolving Iawn sprinkler' Watch leaves cascadeon all sides of their slowly flowing green stems and vines.View leavesfrom up closeand seethat eachone is like a miniafure river delta, a stream of flowing life depositing living green silt upon the web of a growing Plrltag9n' tn mom"e;s of insight, as you practice this way of looking at as vegetation, you iray see all ihese eddies simultaneously

Leaf structuresoften disPlaY vortex streetrhythms.




The pattem of the vortex sbeet appears in the alternating placement of griPPing . suckerson an octoPus'ssP1ral tentacle, as well as the gripping tire treads

vastly nestedwhirlpools within whirlpools of livinq tutP": Ience.Suchself-repiicationis the hallmark of nature's spiral growth gramrnar.

RHYTHMS OF MANIFSTATION of From the contemplation plants,men might be invited to EnquirYs. Mathematical -Nehemiih Grew (1641-1712, plantpltysiologist) English The Bride of Frankenstein's waly shock of white hair grows along the central Path of a vortex street.

There's little doubt that the final years of the twentieth century are a time of rapid global, national, and individual transformation. Rising higher on the spiral of experience, a new world requires a fransformed vision based on cooPeration or partnership with nature, not exploitation. We'1lhave to see nature on her terms, as she is, abandoning the false images that have led to environmental disasters. Most of these delusions are centered around static Pictures, nouns with distracting labels, snapshots, and pigeonholes, such as "greenhouse effect," " ozone depletion " "waste management," and "pollution control'" Instead, we can leam to see nature as verbs, movies, onsoing prccess'and work with these to discover ways to prevent worsening conditions.



help us lnsieht into nature s geometric language will qrasp h"er message.We can begin by learning to Look at and gar;lanis di fferently, any plants, from house Plants to ihe greatiuniuir, to th" gto.ery's'fruits and vegetables, glesand foreits of the planet. We can train ourselvesto vlew energy' of il"ott t ot u" " things" but as patterns moaing a plant grows over a Penoo Have you ever noticed how of davs and weeks or how a flower oPens, or the way new leavesunfurl on a vine? They spiral into existence'resem"energy bline whirlpools. Plants are not "things" but It ii only our slow nervous systemthat fixes sur"t"rit"." takessfatic "snapshots"of plants when the reality facesand energyin mohon' a sPrrais continual change,slow process, path. Practiceobserving plants over time, in whole and m 'cletail, and your vision will gradually shift to conformwith life-facts raiher than the incomplete pictures and clich6s we file within ourselves. As you look at some living vegetation' use your imagination to visualize plants as graphs ot energy in motion. $/hen you have seenplants this way you may want a deeperunderstanding of their rhythms. i pineconeis helpful for this. Look carefulJyat one with ur, uttitod" of seeing anew and you'll notice that each bract-the browru modified leaf that covers the seed ("pine nut")-lies upon a spiral chain of bracts winding {rom po}e to pole of the pinecone. Parallel to this spiral &ain areother spirals l"paraitichies") that also wrap from pole to poie' Eachbract is also part of a spiral that spins in the opposite direction from that of the first spiral, at a different slant' Follow it around. Other spirals lie parallel to it. The point is rhal eachbract lies where two spirals cross.Contemplate the pinecone as the physical manifestation of two oPposite whirlpools. \fhere their lines of iimrltuneously "fi"iti"g manifests. force cross,Iiving matter Onceyou begin to seein this way you may be surprised how obvious the spiral architecture of plants has always at been. Living nature comes about by whirling into mani{estation. Ai fiis! as invisible, Iiving light, it thickens and swirls in energetic lines of force. Upon these lines the more dense substariceof the ancient "world mother goddess" (the four



phases of matter/ or mater) Prccipitates'descending from J iieht to gasto liquid and finally solid expression' he sPlraIS manifestationot the arcriethe material oith" p"itt".ot d1'namic energy \\4rere the lines cross' the lines of "lre woal d/acts precipitaie and become visible' Matter preclPitates *fr".u po.itirr" and negativespiral rhythms synchronize' AIl conifers-spruce, Fir,redwood, giant sequota'nemce the modified flowers we call lo"f- u.J (172V1794)' "yprerriprodt .o""t. ffl"'6*iss naturalist Charles Bonnet t ,o*. -airrty for his studies of insect metamorphosis' coliabonted, n1754with the mathematician/ artist G' L' CalanJrinl fo. tfre first modem mathematical study of branch and l"uf ur.ung"-"ttts, beginning with the pinecone' They .oi""a tf't""*o.a phytloiaxis(Greek for "leaf arrangement") when they noticed ire cone's definite structde of opposite-'-wh; spirminq spirals o{ bracts. of tiey discoveredwas that counting rhe nu-mber parallelspiralsn eachdirection (not the numbers of bracts!) will inevitably yield consecutive numbers along lne Fibonacci series, often five, eight, or thirteen' Their ratio of clockwise to counterclockwise spirals is an approximation of O, the golden mean.This ratio tells us that the conehas an inherent ialance and harmony. If the numbers are low on the the sequence approximation to O will be crude Higher produce a'ratio closer to the mathematical ideal' numbJrs is I brurwoluf of grass Each plant rnanifests the approximation that best suits itg neeos. no lessthan lhe Countless examples exhibit this identical structure, journeywork thestars of although they display different numbers along the . . . and therunning Fibonaccisequence. adornthe zoould blackberry Modem 6otanical thought on this subject is that the cone simplv accumul atesbracts, mechanicallyemerging wherever parlorsof heaoen. they fit. But deeper insight will show that the bracts are pre-Walt \{hitrnan cipitating uporthe invisible lines of force where opposite W" are seeing the manifestation of the rhythm spiruls "-"i. fieId. o'f urr un"rgy Just as the diminishing golden rectan-gle -oi whirling squares in its wake, the power of life left a trail trail of living matter in its wake as it whirls through leaves a the world. Plants are the visible clothing of the creating Drocess. Betteryet, observeliving vegConsidertheseexamples. and grocery are libraries of greenhouse., etation. The garderg



(,',i',i t t ,' I

l,t,i t tt


and blackberriesare really many fertilPineapples,raspberries, one fruit' *--fn" i^a J*ti"t ,r,j, r,ave spirally accumulatedto form fruit with its seedsspiraling ti..*U".ry is an inside-out around the outside. " " kemels Co^, *tt"ut, und many blossomspackagetheir seed kernelsseemto form straight rows h e{ficientwhi-rlpools.Com's ("straight rows"), but take a closerlook: the ilTed orthostichiis kemels spiral around and around the cob (eye)' The ihoms, spikes, barbs, and needles of cacti and.other . rn olants wind around in spirals. Thesewere once central verns water in harsh environieaves, which diminished to conserve rnents,like pine needles.

The thoms of the rose wiaP sPirals, around in oPPosite revealing the subtle energY_ ot pattern of the archetyPes the creatingProcess.

the Fibonacci rhythms of nature's creating process' Verify that each seed, huit, and leaf manifests itself where two lines of force cross.If you wish, count the oDDosite-spirming n'u.r,butt oi paral'ielspirals in eachdirection and use a calculator to de-terrnineliow close each is to representing the O ideal. The architecture of upright plants shows that they put out leavesin spirals around the stem, like a spiral staircase



Within the Plant's unfolded stem, its "Ie#-trace" Pattem feeds altemating leaves in a Fibonacci r\thm.

counterclockwinding in either direction, clockwise and on pages 170-171 slopes.The illustrations wl" "t"airr"."itarrangementsof some familiar plants anc the leaf show fittl"g their lJaves/spirals ratio and characteristic t ""t, U"t*;"" consecutivefeaves,moving both clockwise ,.gi" and counterclockwise. Within eachplant's stern are veins, or "leaf traces'",noura istring the leaves.Eachvein feeds a chain of leavesalong not in its Pattern'J* l11t: furtiSU"t patt"t", passingthose and lower leavestecl Dy me ters of Ieavesbetween upper comprise one-'1eaf cycle" reminding us of the r* ""irr between waves of the sea' wavelength of As vJur eve waiks up and around the spiral staircase the number of leavesin one Laves, vot will discovei that leaf cvJle, that is, to the next leaf fed by that vein' is a Fibonacci number. Also, the number of spirals you tum to get to the next leaf cycle will alsobe a Fibonaccinumber' Each of the world's 350,000known plant speciescan be characterized by the ratio of "leaves-to-spirals" in one leaf cvcie. The hisher the numbers are on the Fibonacci sequence ui-rdth" -o.6 ot"cise the ratio is to (D,the more complex the plant (e.g., fems two+o-one ratio is crude, sunflowers is finer). By multiplying the inverse ";gtrty-.tltt"+o-fifty-five

The ideal angle between consecutive leaves of a plant sem from above,the "sun's-eyeview," provides minimum overlap'



"The leaf always tums its upper side towards the sky so that it may be better able to receive the dew over its whole surface; and these leaves are arranged on the plants in such a way that one covers arrother as little as possible. This altemation provides open spacesthrough which the sun and air may penetrate. The arrangement is such that drops from the fust leaf fall on the fourth reaf in some casesand on the "r\:r:;rrf;;rrnci k. 1s00)

of this ratio by 360 degrees (to get a measurement of less you will discoverthe anglebetweenmeathan 360degrees), surement of consecutive leaves as they emerge from the stalk that is characteristicof the species.The ideal angle is approximately 137.5degrees(= 1/(D x 360),or 222.5degrees (its cornplement), measuring in the opposite direction. On a clock, the angle would be made by the hands at three minutes to four o'clock. As Leonardo da Vinci noticed, there are supreme advantages to this spiraling arrangement. The more precise the O ratio is, the less overlap of leaves there is so that each leaf gets the most sunlight and the ieast shade. Plants seem to configure with the approximation to the ideal that best provides their needsfor light, atmosphere, and balance.Look at plants from above, from the sun's-eye view, and notice that their spiral configuration allows you to see each leaf. The design also producesefficiently structured water collectors, solar water heaters and photovoltaic collectors, which are also roles of olants. The golden spiral's geometric characteristic of self-simiiarity of part and whole guarantees that while the plant grows in size and weight it maintains the samebalance, stability, and center of gravity.

The green plant is the visible part of a spiraling energy field.

A rhirg -oy ,ndurein nature it is duly if proportioned its to necessity. -Fra Luca Pacioli

170 Fibonacci fraiction of leaves '---1-'=



z 1





'@'@ 4 1 240" At gl" between consecutive leaves




Leaf arrangemeru: Plant as spiraLing field

Grasses,rice, corn, wheat, sugar, barnboo, fems, palm fronds, lime, iry, elm, basswood, lindery ash, horse chesbnut,sycamore, maPle, dogwood

Beech,hazef Potato "eYes"



5 2 \

q 5

,(,M -\Y" 1



9. 1 216"




Apple cherry apricot, oak, cypress, poplar

Holly, pear, spruce, various beans



BUILD A PLANT REPLICA of the structural With someskill it's possibleto build models paper' scissors' a plants' You'll need stiff tf"rv,i-t .f in a base "'t".y tf."t"Ut"itr, urra an upright rod rnounted "a'""if, ?thiscould be a pencil in a ball of clay)' First, createbr transfer the following spirat q":*::l:" at ideal rJl'cfirm paper or cardboard Points along it are consecuJ"niJ. i"t"-urt from each other' Note how each pentagonalpattem'.cut a whirting ;;il;" ;;;.ompose and lift the eye into three dimensions' tacKng out the spiral ,h;;; t" that the s#ip spiralsaround it Tapetheboti;; or taPemerr tom to the base.Cut out PaPerleavesand Slue

haoe Where plants fiae,a fold Patterns of considerationtheirsouls of For is in place. Patterns in fiae appear theregular the and solids, soinaolae the ratiocalled golden results which section, ftom series a self-deaeloPing of that is an image the in facuttyofproPagation Thusthe plants. flower the carries authentic of flag the thist'aculty, Pentagon' -Joharmes KePler




The.numberof stemsonto the rod behind eachof the points ot sPtals' *leavesis a Fibonaccinumbet as is the number illustration can be ;; oiu.t e*a-ples in the previous on tne replicaiei in threedimensionsby replacingthe Points paper spiral with those of the plant's characterlstlc.angle' s eye' '.--X usins a protractor measuring out from the sPiral and thread dra-wnthrough consecutivepoints "JJf" view' wlll crealong the spiral,when seenfrom the sun's-ey-e showing star,the flag o{ life and excellence' "i" "h"t#"tn are at tle heart of living ,lt"i it" ot"-.iples of the P-entad structures.

F or theworldis not but or painted adorned, is from thebeginning not and beautiful; Godhas beautifuI some tnade is BeautY the things,but of creator theunioerse -Ra$h Waldo Emerson

THE CALM "I" OF THE STORM We've seenhow the archetypalprinciples of number manias fest themselves forms around us Circlesand spheres'rrlspirals,and the five PlatonicvoIpentagons, angles,squares, principles that shape the world' \Alhen we um"es.epiresent see stars and spirals we know that there is life and excellence,motion, bilance, and transformation But the universe is noi like a roomfitted with geomekic shapes Theseforms What of arethe shapeo/the universe,the shapes space.itself' forces and ur",i touch are the visible expression of *" """ principles we can train ourselves to understand' For the iake oi solving our environmental and social problems, we can, and must, learn to seewith new eyesThe lesson of the spiral is that every "thing" is not a noun a bliit a process, d1'namic " energy event'" The world resembles a whirlpool of transformationwith which we can cooPerate for our benefit. We can also use our experience to leatn to seeoulseloes differentiy. A glance at the human body shows that-instruc-tutt"tiot we carry the pentagonal flag of life and ture and Like other living forms, our body's ideal proporexcellence. relations grow in accordancewith the goldenrnean, the <D tionship weaving balance, harmony, and beauty among its parts. ihis same accumulative proportion- is found in the physical structure of living sea,land, and sky creaturesand in ihe paths of atoms and galactic clusters' It is stirring to realize that each of the physical structures of the universe is

The miracleis that the a created Partof uniaerse itselfto study therestof it, and that this Partin studyingitselffinds the in restof theuni9erse its natural inner o7fin realities. -John C. LinY



lacob'sLadder,a watercolor by William Blake, is depicted as a halsforming spiral shetchhg between earth and heaven.

a repackagingof the others.We hold the proportions of the solar svstEm"inour hands, face, and whole body' One can fold an outline of the human body differently and createthe a proportions of a starfish and seashell, rose,and the Mi-lky entire universe is a great' living' Wav ealaxv. Perhaps the intellieent beine we iust don't recognize,in the same way that oie cell in"your-body doesn't susPectthe existenceof the person reading this or even have the ability to comprehend vour possibility. -the world outside ourselvesto the subSv turnins from tle world widin, we move from the symbolic to the sacred' principle of the Pentad,is In symbolic geometry the essential expressedby the property of selfIl fe and "reieneration," similarity. Inwardly, this characteristic indicates the possibllity of ipirituat regeneration, rebirth from the human to the divine as sought by every one of history's religions' Methods for attaining it differ, but curiously they share the spiral asa universal s)'mbol of spiritual transformation,just as spirals indicate the transformation of forces in nature. kv the Bible the Deity speaks to prophets from the same whirlwinds on which Native American initiates, like those in many other'cultures, were carried to "heaven." The approach to heaverL not a iocation in the clouds but a s)lrnb^oiof regeneration to conscious identity with the Higher Self within, is never depicted as a straight line' Neither is the descentto hell, as Dante tells us from his downward spiral tour. Craftspeople frorn aricient Eg1pt, Babylon, China, Africa, Europe, India, and Pol;mesia and great artists from Brueghel to Blake have s)'rnbolized the spirifual path as a spiral. AII people seem to undelstand this intuitively. Even the rnakers of popular movies feel it. On her dream joumey, Dorothy is transported first from Kansas to Munchkin Land on a spiraling "cyclone" and travels from there to Oz via the Yellow Brick Road, which begins, you will notice, at the eye of a golden spiral expanding ever outward to the Emerald City. With its inherent harmony, the spiral is a metaphor for our inevitable transformation within. This inner spiral is not the physical spiral of our body, the embryo and ears, hair and heart, fists and fingerprints, but rather a psychic one,



The Yellow Brick Road, symbol of transformation between Munchkin Land and Oz, begins as a golden spiral.

and beckoning us as a s)'rnbol of consciousness the way we live. Our own inner spiral process whirls us into the mystery of the infinite within us. The spiral shows us that we are comprised of a continuum, not fragments, as we ordinarily appear to ourselves. A clue waits coiled in the spiral's principles of self-replication, self-accumulation, self-recurrence, and self-similarif. The recurring key word is "self." The message of the spiral is growth and the transformation of our Self. At the center of our Self, deep within out consciousness, is a calm "I." Like the calm "eye" within a storm, our center is untouched by psychological turbulence. Peaceful, it observes all from the vantage of wisdom. Placid, it is r;nmoved by the turbulent weather of the surroundi.g P"yche. \,Vhen you re feeling connected with your center it seemsvery farniliar. It feels like the Self you know best, Iike who and what you know your Self to be, calm in knowing

mark It is an eoer-fixed and that looks tanpests on shaken. is neaer -Wi.lliam Shakespeare



of Wi*routlookingout thewindow,onemaYsee theway of heaum. -Lao -tz'r (c. 604-5318.c., Chinese Taoist philosoPher)

being without thinking. To be centered is not the same as 6r selfish. lnstead, it is identity with the "Jt-""ttt"."a" deep, divine power that motivates us' 'Considerl spiraling top spinning "centered" on its calm eve.As soon as it tilts slightly, the top goesott centerano.rts outh rarid"nt, spiraling away from its center ol stabllltyit iwav from its centeroigravity. The further from its center the center createsthe moves, the more it wobbles.Leaving trail of the spiral, an attempt to regain balance . The center within us is the truth of our fugher Selt' tne reality deepest within which each of us calls "I'" Each time we lose sight of the "I" we spiral toward the periphery' We of we aooeartoieave the centerbecause feel a sense separain our lives We lose ow vantage point of the ti'oir somewhere center when we artificialiy divide our Self and try to look 4t we it, think about it, want it, Sraspat it. \ 4nen leaveourcentel, we are simPly identifying with other.Possibilities ot our Seit aistractea Uy the turbulent realm of the four elements' And when we do, we rniss the peace of the calm "I" and yeam to retum to it, to transform our selvesback to identity ;ith it. Identity with our center is always possible, as a wobbling top can sometimes be helped to regain balance.' ihe^further we wander from our cakn center, the calm "I," the greaterour experienceo{ psychological-turbulence' The motion of stepping away createsresistance,like moving an oar in water, leJving in its wake vortex streets o{ thought and emotion, desire and urge. Drifting from our center takes us away from the calm by creating resistance, resulting in the sometimes stormy turbulence of our psyche' Like water and air around us, fluid emotions and airy intellect within us moves in spirals. Like the golden spiral of nature, we grow by experiencing resistance and overcoming obstacles, Idlustlng our path as we go. We know we are off-center when we drift into identity with the clamor of sensationsthe dramatic whirlpools of passion and fear and anger, dust devils of desire, swelling waves of emotion-and when we are kept awake at night by winds of thought' It'i a natural part of our growth processperiodically to leave our centet and come back to it, iust aswe must expand and contract to breathe. The creative pulse of the Dyad between center and turbulence is part of our growing



to Drocess.The rhythmic interplay between the impulse that is iivide ourselvei fto- orr. center and the yeaming in expressed ow longing to retum is found in the experience 'constantly searching ior and find ing meaning in our lives' of The interpliy betweei these two forcesmanifests itself in dheworld^.as'a curve. That's why the transformative path is characterized by a sPiral and not another shape' A soira.lpulsesall ways simultaneouslyand yet remains .o.,rturit in its properties throughout change' By being aware of our inner motions and learning to observe ourselves, we can find the calm eye or calm "I" within the weather surrounding our centerof awareness' When we gtow, *e arrive at the same qllce -where ye were before, bit we arrive more experienced, higher on the soiral.

There is nothing what excePt is pleasurable with the in harmony

of dePths our utmost dioine nature. -1366' -Heinrich Suso(1300? GermanmYstic)



$ttucttttE'Iu Itdet l|rxnn Srx And God saw everything that he had made, and behold, it was very good. And it was evening and it was moming, th: tYt . day. ihus were finished the heavens and the earth, and all their host' -GenesisL:3'1,2:7 Six days shalt thou 1abor,and do all thy work. ' -Exodus 20:9 God has established nothing without geornetric beauty which was not bound beforehand by some law of necessity. -lohannes KElcr All the pictures which sciencenow draws of nature and which alone seem capable of according with observational fact are mathematical pictures . . . From the intrinsic evidence of his creation, the Greit Architect of the Universe now begins to aPPear as a pure mathematician. astronomer, -Sir lam* H. Jeans (1877-1946, Englishphysicist, and wviter) Six of one, half a dozen of the other. -EoIk saYing

Birth of the Hexagon. fla



SIX APPEAL iourney to ten' We've now passedthe halfway Point on or-rr "Six" is the-fou rth actual number according to the mathematical philosophers (who saw the birth of numbers at "tt'ree" and ending at "nine"). We could probably be understood around the ivorld and throughout history becauseof the similar pronunciation of its name in eg)"tian (sls), e:fSanskrit (sas),Hebrew (sesh),Arabic (sittQ, Cel.t'Lc ian (sissa), Spanish Irish (se),Latin (sex),Ita1ian(sel),German (secfts), Russian (sestfl,English Grx), Danish (sels),'Fiench (sees), (sek), and other languages. The archetlpe of sixness.was called Hexad by the Greeks, in whose language the sibilant sound of "s" becamethe asPirated"h." Whenever we encounter the Hexad in countless hexagonal (six-sided) natural phenomena and human designs, from beehives to faucet handles, we can be sure that certain principles are being called into play' By investigating the lrithmetic properties of the number six and the gegmetd.c properties of tl't" he*ugon, we'll find that the Hexad is intimate with both Monad and Triad, the circle and triangle, unity and hinity, wholeness and balanced structue, apPlying their principles in an advanced way. Hexagons contain a message that efficient structule, function, and order are occurring. But these three separate words represent a unity; the qualities they name are always integrated, never separate,and must efst simultaneously or not at all, like BorromeanRings.We'lI refer to Hexad's principle as "structure-function-o1der" and "space-powerhave recognizedthat separatingspaceand time." Scientists time was incorrect, so they call it sPace-time.But there is a third factor, power, which must be included. Every whole Hexad event occursat the intersectionof thesethree aspects' represents their framework. Consider any whole event or "thing" and recognize that it has a spatial sffucture, inherent power, and duration in time. M/hat else is there? Structure-function-order. Ponder the words separately, then feel them together as one concePt, as a simultaneous. mutuaily reinforcing whole. Work into your "feel-knowing" that the siruciure of any event determines how it can func-



tion, that its functioning takes place in an orderly sequence, and that this order of unfolding determines what the structure must be. Sixfold phenomena serve as the visual representation of the Hexad's imprint of structure-function-order. ln surprising ways, multiples of six, particularly twelve, thirty-six, and sixty, also emerge from the principles of the Hexad as a "natural" framework in mathematics, nature, the syrnbolic arts, and everyday affairs. The dodecahedron's dozen faces,the traditional representation of quintessential "heaven," was associated with the twelve constellations of the ancientzodiac.The story of the plocessof the sun'siourney through the cycle of the year told through the s).'rnbolism of the zodiac has endured for countlesscenturiesas an archetypal tale of twelvefold progression on many levels. The story of the starry skies was purposely mirrored in terrestrial affairs as the many twelve-part epic myths of solar heroes (as the twelve ordeals of Gilgamesh and the twelve labors of Hercules) and in a surprising number of religions (particularly those having twelve disciples around a central figure, modeling the zodiac around the sun). Corresponding twelvefold structure-function-order in architectural design and symbolism was essential to the astronomically aligned temples from Stonehenge to the Goihic cathedrals and Native American tepees.It's well known that whole societiesfrom the Twelve Tribes of lsrael to Solon's Athenian society and the Chinese and European courts were organized according to the duodecimal cosmic ideal. Even the proportions of everyday weights and measures were based on the natural twelveness evident in mathematics, which makes subdivisions and multiples easier (since twelve and sixty have so many divisors). This supreme structure-fi.mction-order of archetlpal twelveness allowed ancient societies to move in harmony with tlLeheavens and to cooperate with nature around them. Embedded in these cosmic and terrestrial structures and stories were allegories known by every ancient initiate to disguise a teaching about our own inner archetypal structure-function-order. Knowledge of twelve-step processes was considered useful for the conscious development of a perfectedindividual, whom the GreekscalledH; Nikln,The Conqueror. He was represented as the Sun God surrounded


by a pantheon of twelve spiritual attributes. Traditionally bom during the winter darkness, dying and resurrecting in springtime, the sun's annual journey around the zodiacal belt symbolized the process of spiritual regeneration taught by the ancients all over the world. The Hexad's principle of unified structure-functionorder wiII allow us to incorporate in our universe structures from snowflakes to the starry zodiac. Have your outer and inner geometer's tools ready, and we will begin as near as we can get to the Hexad's archetypal birth.

A PERFECTSIX The ancient mathematical philosophers began to understand the principles of any numerical archetype by first looking at its arithmetic and geometric properties. The unique properties of sb<ness will help us gain insight into the Hexad's principle of structure-fi;nction-order. The Hexad has unique number properties. Six is the first of orily two terms within the Dekad composed by the multiplication oI two differmt factors (other thm It shares "tti'1. ihis athibute onJywith the completed Dekad (seechapter ten):

1=1x1 1 - 1 v 1

3=1x3 4=2x2

6=2x3 7= 1 , x 7 8= 2 x 2 x 2 9= 3 x 3

5=1x5 10=2x5 Because six is a doubling of three it partakes of the Tiiad's principle of balanced structure. The Hexad is in many ways rooted in the Triad. The ancient mathematical philosophers, viewing numbers as coilections,of Moruds arranged into geometric shapes, saw six as the tlrird of low triangular ntilnbers within thi Dekad (L,3,6, 10). They also noticed that six is both the sum and product of its divisors, the first three terms (1,,2, and 3). Thus, to the ancients, represented parents(1 and 2) of all numbers six the with their firstbom (3), thus making a completed whole. They called it the Perfection of piarts,since six is the only



nurnbersSrow as Triangular DY their bottom row increases consecutive terms of the counting sequence'

o 1

oo o ooa oooo oo 3 6


o aa ooa 10

of the same three number that is both the sum and Product

'i",-"e"*. q:Ti i"", r" f : :: l"l'J trfff "#fi H: Jlf set of three integerswnel or4lv sum of the othertwo, "-";;,h; t t"iguedby the interplayof two and4""'T*

"",i;;;;t ;fiili";;"h"tt

and sawin both thePentad Hexad by,the bothformed thev're since fi:;;;il;;""'""rriage" even and

m"tl"'Tlti*1 = s anaz x s :6' iheancient differ-

;f ;;;'pi;t '*a a femaleierm (two is considered male)' is *ale term (three odd andconsidered i"-uiul " Plogonly five and six generate parents, etrJtlL Uiotogical

"-;tk" ^iri'r"t

The Hexad is sometimes spnbolized bY the '?ythagorean triangle," or '3:4-5 right hiangle," made bv the ancient method using a"twelve-knotted roPe. It disfrom one olavs the sequence io six: one right angle,two uneoual angles,sidesof three,four, ind five, enclosing an area of six square units.

in-a five (5 x 5 = 25,5 x 5 x 5 125' = end a srx(o x o ""a fikewise, the powersof six necessarily m (aside from i?. ^\'&[ :- ile, andsoon)' No otheinumbers of properties all numthe -*erates and shares "iir.'*f""n tf'rJubility to generate suchmodelsof themsIves' U"t.i flu"" in tlis self-replication{orms' ;ili';;iJ".;;esses ntorcrng in selt-rer its self-similarity The Hexad "*pr""""t dubbed the Form of Thus it was il.;;;:tu".-{;order' Anvil by thePythagoreans' Unwearied lor- u.tatn" " then,the Hexadrestson a stabletounIn ai1ways, dation whosepaits aremutually suPPortins' THE BIRTH OF THE HEXAOON will showus how,the constructinga hexagon Geometrically manifestsin universal Flexud'slntemal arithmetii structrire desieal. """ known for 3i.." antiquity, various methodshave been sorneof which areshownhere' Ttre constructinghexagons,

ds:.kes; thatis,selimultiplyingPsTt:"f on)' l'l: =



hexagon is arnong the easiest forrns to consffuct, emerging natuially from the circle, the Monad' piscis' (1) First, conMethod one: Birth through the uesica piscls. (2) Uie the straightedge to draw a line sftlucta aesica cormecting the circles' centers, extending them to meet each circle's ciicumference. (3) Place the compass Point at one end of the line and turn an identical circle to create a mil(4) rorirgaesicapiscls. Revealthe hexagonby corurectingthe around the central circle, that is, the two centers six o6ints and four points created where the circles cross' Method two: "Walk" the radius around the circle' An even simpler method for constructing a hexagon, the one you proiablv leamed in school involves "walking" the comPass 'open to i circle's radius around its circumference. (1) Center yburself and tum a circle. Keep the tomPass open to the size of the radius, (2) Put its Point anywhere on the circumference and walk it around the circle, marking each point where the pencil crossesit. If you're careful, the sixth step will end precisely where the first step began. (3) Connect the points to form a hexagon, and then corrrect altemate points



Method one; Birth through the auica piscis.

Method two: "Walking" a circle's radius around its cirThe hexagon is cum{erence.. an exteriorization of the circle's radius. Intemally, it enfolds six Tiiads.



Method three: Six identical circles always surround one.

to form two interlaced equilateral triangles known as a "hexagram" star. Ev"eryhexagon is a statement about the relationshiP of a circle's circumf"erenceto its radius. A hexagon is the circle's radius externalized, iis circumference internalized ln other words, each one of a hexagon's sides is equal to its own radius and to the radiris of the circle circumscribing it' Method three: Six circlesaround one.It's a well-known geometric fac! although no one knows why it happens, that six identical circles fit precisely, or "kiss," around a central circle, eachtouching the center circle and two others. Prove this by arranging six similar coins, cylindrical glasses/or oranges arormd a central one. To the ancients this arrangement represented the weel with six days of activity around a cenhal Sabbathday of rest, just as the compass rests at the central still Point. Reproduce on Ancient Pottern IJse the six cornersof a hexagoninscribedwithin a circle as the centers of six identical circles to produce a lovely decoration used from time immemorial upon pottery, tiles, wallpaper, and fabric. Construct a hexagon using method one. Extend the hexagonal pattern by continuing to place the compasspoint wherever two circlescross.This pattern has been used as a tool for self-observation. Gaze at it a while. Watch how various symmetries stand out and transform before your eyes.Theseshifts are a result of fluctuations of our own nervous system.

Tile patiem (Pompeii).

Islamic tile and ornament pattem based on the hexagory called a "true lover's knot."

Lincoln Cathedral East Window hacery c. 1260.



Construct on EschertgPe Design the The Dutch artist M. C. Escher (1898-L972) applied with images ancientknowledge of hexagonaltiling pattems from nature and fantasy to create intriguing deslSns'..You can, too. Just follow theie directions as you look at the illustration: subdividea hexagWith a light penciland straightedge, (1)Startat any singletrional srid inio aiieta of triangles. ansle'and draw a curve between two of its points' You can ,r.J the crrve in this illustration to create whirling birds, or at vou can makeup your own curve and be surPrised what {ror,..*ut". Ql "Pivot" the curve at one corner of the triangle 'and draw the exact same curve along the sides of six connecting triangles that make a hexagon. (3) On the original triangi-e's reniaining side draw a different curve, one whose halvi reflect eachother in mirror s}'mmetry around the center of this side. (4) Repeat this curve around the outside of the hexagon. (5) Now take a moment to design the bird, using simple lines for its spine, wings, and tail feathers (6) Aipply this design to the six birds of the hexagory alternating-dark and light, or with coiors, to distinguish them(4 Repeat this processin adjacent hexagons,noting how Pennsylvania Dutch "hex" sign.

A}litr:.d.uyantra, or medita-



threehexasonsmesh,with tfuee light (or dark) tails meeting at a comm6n point, to fill the sky completely with interlockins bfuds,leavinq no gaPs. " With practice"andly seeingthe resultsof different initial yi., .utt get qui[e good at designinginterestingtiling "r*u", patterns.

Construct a hexagon.You'll need two identical striPs of paper.eachas wide as half the hexagon. LaY each striP on the hexagon and fold it (as shown) upon the hexagonal guide. CarefullY join the two folded strips so theY lilk. This knot is known to sailors for its ability to link two ropes together with great strength. Simply stated in the language of natural design, the fust principle of the Hexad is efficienf economical strucfure.

N rr$2,



Tie a Hexagonal Knot.

The two ends of the Nile symbolically unite ancient Egypt wiih a hexagonal knot. The whole picture was designedin a hexagonwhose diameter is the height of the pole. Connectingits corners and midpoints shows the figures aligning with, and Partitioned by, the hexagonal geometry.



The hexagram star is an ancient s).rnbot agnell8 [{orldwide religion aird myth. [t is sometimescalled the Sealof Solornon (wlich Moslem legend recountshim using to caPturemagically )iinn, or " serne,";ature spirits). As the reveredsymbol of the And to Hindus' it ;lwistr peJpte,it is calledihe Shield of David' the hexagram star is is the Mar[ of Vishnu. In some traditions, the great seal of initiates, signifying rising aspiration from below metly the descent of grace from above.

LISTENINGTO STRUCTURE its The hexagon holds within it three distances:the length _of (equalto its own radius), its diagonal (twice the radius), side and the distance between altemate comers (root tluee [= L.732.. .l times the radius). We can seein the illustration that the three lengths form the sides of a right triangle having sides one, two, and root three within the hexagon' These are the numbers associatedwith the circle, line, and triangle. A hexagon makes visible the relationships between the archetypal principles of Monad, Dyad, and Triad. The relationships among these numbers are abstract but can be approached in another way. We can make mathematicai relationships sensible by hearing thern. Number relationships can be made audible through music. Try this: construct a large hexagon op a flat piece of wood and hammer nails partly in at the thre comers of the inner right triangle as shown. Tightly wrap a single guitar string, piano wire, or

A curious fact is that a circle constructed within the inner hexagon of a hexagram star has a circum-ferenceexact$ half that of the large outer circle. Verify this by measuring both with a piece of string.



@{ Nuts and bolts

nylon fishing line tightly around the nails, keeping equai tension between them, and tie it tightly' Then simply pluck the lengths and listen. Since the tension is equa1,only the length of each string determines its pitch. The shorter the string the higher the tone. The three form a related sequence of tones, a harrnony characteristic of the hexagon \44rat is important in this experiment is not each tone other.We canhear the relato itself but their relationships each tionships by ptucking the strings sequentially and then sirnultaneously in different combinations. They compose a harmony, basedon the "square root of three" (= 1.732.. . to 1), the ratio within an equilateral triangle, the Triad, the archetypeof balance.

Glass turnbler

HUMAN HEXAoONS Hexagons appear endlessly in human inventions, providing greatest structure-function-order, that is, maximum efficiency of materials, labor, and time, by using straight lines to approximate the efficient spaceof the Monad, a circle. Its relationship with the Triad gives this form the great strength of three-comer joints. Designers need not know anything about the mathematics of the hexagon to discover that it works. Some hexagons stand alone, and some tessellate, or tile, in repeatingpattems. The single hexagon works in situations requiring strength and siability, using the fewest materials and straight lines. We know its effectiveness when we grip a faucet handle or tighten "hex nuts." A hexagon provides stability at the base of a fumbler and in the six central ribs of a woven basket. Bicyde racers know that birycle wheels are most efficient in terms of having the least weight and wind resistance yet the greatest strength if the number of spokes is a multiple of sir; usually thirty-six. Any design that requires a circle's efficiency but is lirnited to skaight lines, Likethe light-limiting diaphragm of a camera lens, will somehow make use of the hexagon. Cfucular umbrelias, caps, and parachutes make the best use of material with the least weight when they consist of six sections, or a multipie of six.

Center of a woven basket

Faucet handle



HummingmuitymaY . aariousinoentions . . make anY but it will neoerdeoise morebeautiful , inuentions nor nor moresiffiple, more thanNature to thepurpose in because her does; nothingis inoentions wantingand nothingis . suPeffluous -Leonardo da Vinci

Many famfiar items rely on hexagonal structure.

fIEXAGONAL PACKAGING hexagons arrange in repeating Patterns they extend \,1y'hen the pattern of "six-around-one." In this way the circles, and the hexagons formed by connecting their centers,are said to packed in a hexagCircles,cylinders, and spheres tessellate. onal pattem fiil spacemore efficiently than in a square pattem, getting more into the same amount of space.Engineers and designers study this principle, but most people learn it at the rnarket by observing the way fruit stacks in sixaround-onetessellations. But when circles tessellate they leave curved triangular gaps between them, Wasted space is little problem when stacking appies, but more efficient space-filling requires that the circlesbe convertedto hexagons,leaving no gaps at all. Hexagonal tessellation extends the circle's principle of "equality in all directions" in a baianced sharing of space, materials, time, and energy. It creates three-comer 120degree joints, which provide the triangle's strength and to structural excellence the package(seechapterthree). I

V* Hexagonal packaging incorporates more circles in the same area than square packing.

t90 Hexagonal tiling leaves no . gaps. The efficient Pattem ot six-arornd-one is so conunon that we hardlY notice its familiar aPPearance.


sffi Computer chiP circuit

make T h, brrt carpenterc chiPs. the fewest -German Proverb


191 Central marketPlaces.Studies find that older cities around the world seem PurPosefullY arranged in a hexagonal lattice, a pattem that economizes market centers, decreasescomPetitioru minimizes transportation distances,and maximizes admirristrative control of outlying areas.





Once you begin to notice hexagonal tessellations you'll seethem endlesslY. Before you read further, construct (or doodle) some hexagonal tlssellations using circles or hexagons'-Get a feel for their pattem of exPansiveness,efficiency, and cohesive strength in close-packing.

HEXAGONS,NATURALLY While pentagonal symmetry is only found in living structures, the Hexad manilests naturally in both living and nonliving forms. All natural forms .are forces made visible. When we encounter six-sided, six-angled, or six-pointed crystals, plants, and animals we know that the Hexad is underlying an efficient structure-function-order. Its threejoints and close-Packingarrangement comer 12O-degree ensure an efficiency of materials, time, energy, and strength. The tessellating hexagonal pattern is common in the realm of molecules and crystals. Molecules q-pically occupy two thirds to three fourths of the crystal; the rest is emPty space- A close-packed arrangement uses minirnum space and best balances the athactive and repulsive forces to achieve the minimal total expenditure of energy. A knowledge of archetypal geometric Patterns is the basis for the

192 Observe how a froth of bubbles in the kitchen sink eventually settles into six-aroundone arrangements as tne molecules strive to balance forces with minimum surface tension. The fleible rePtilethe skin pattem exPresses ol torces ideal energYPattem in balance. The facets oi a fly's eye are arranged in close-packedhexagonal arrangement.


Jabrication ol modem sciencedf crystal engineering' the ano molecular crystals with unusual oPtical' electronlc' '^'*il"maeneticproperties. see arrangementsof atoms.and mole"-uiJr'"*ily their Pattem on a larger scalern a rrotn cules,but we can see of butbles, in snowflakes,and in quartz crystals' Sincenature traffics in efficiency,hexagonaltesse'llahon oi wooo recursendlesslyin nature' The molecular structure larger scaleaswaterit u f,Jr,agonal web, seenon a ""it"tot" tt va-dietyon). The star coral configuresa hexag' ""i "f*"1 *d tube coral grow by leaving their cylindri;;rl "?;i"io'", packed together in hexagonalgrouPs l\once cal skeletons n#ble hexalonal tiling i*the arrangement of ;h;;; on scales a fish or in the skin of a repiile' Iike the chau'l-mall .1 tttiglt,"' armor. A tortoiseshell, with its hexagonal ""f the shape of minimum energy and maxiolutur, drJt*"s -mum strensth. "human body individual cells join in a closett", tt't" oacked hexaqonal pattern to act in concert' The so-called that occurs throughout our bodies is hexagit ip"a -uti" sucn onallv packed and available for voluntary movement'



t: the along page *'d'-T?: aswhenyou moveyoureyes lunC in recurs human if:j: words'ThefishingnetPaftem dloxloe for oxygen-carbon maximizing the passageways tansfer in each of our breatns' only encountered in living Single hexagons are com

.."fi""i;;,i"";:"A ;?1;;iil;8onal tai"t ^", ti**i,

insects' ahd some sPonges'

sliceof carrotuttdtL" top "f ? 8r-",",1ff1"radrolarra' structwe'asdo microscopic

a Human lung alveoli form nei. hexagonal

toP' like the top of a tomato and A carrot slice and $een PePPer reveal their hexagons' o,ft"t f*io *a "eigetabies,

Dothnot natureitseli teach You? -I Corinthians 11:14

walks ugon tfe All tme insectshavesix legs This water strider asif on a net' lifting altemate ilt!.-ro.tu""'" *ot"cular iension Iegsin stabletetrahedralfashion'



Who ,orordthenucleus, before fell, into six horns it

of ice? -Joharures Kepler


appearance- crystalline snowflakes is why scientists of con{e slder water a mineral. As the temperature drops, H2O molecules vibratemore slowly, slow enough for electriccharg6s witlrirr moleculeto attract other moleculesand tighten into a hexagonal, "; close-packed arrangement.More moleculis build upon the'seeJ' pattem to become beautiful snowflakes, blanketing'the world with sixfold svmmetrv.


Close-packed silicon atoms arrange in a hexagonal pattem like ice and grow as a six-sided quartz crystal. Due to its electromechanical, or piezoelectric, properties, pressure on quartz yields electricity, while electric current put into quartz lrom a battery produces regular mechanical pulses, hence, the "tick" of the "quartz watch-,, Even in this most modem sensethe Hexad is associated with order and timekeeping.

The graphite in pencils is made not of lead but carbory the same element that comprises diamonds. But diamonds have tetrah;_dral architecture, making them nearly indestructible,white

ff"X#: fl;:"r

sheets sbde thepen_ rhat trow in hexagonal off



CH: H Uranium hexafluoride VitaminC


Six water molecules form the core of each snowflake.

Cyclo-octa-deca-nona-ene, an aromatic fragrance


the The discovery of the benzene ring (C6,H6), basic structure of organic chemistry, came to the German chemist Friedrich Kekule after he dreamed of the orraboros,the circular snake biting its own tail. The Hexad's principles configure wherever efficient structure-function-order is required. Molecular chemists and biologists find them in irurumerable familar substances including steroids (cortisone and the male and female honnones testosterone and estrogen), cholesterol, benzene, TNT, vitamins C and D, uranium hexafluoride (nuclear fuel), aspirin , sugar, the antibiotic Terramycin, and pencil graphite. The atomic structure of cyclo-octa-deca-nona-ene,an aromatic fragrance, reminds us of Islamic pattems and beehive

of [The existence pmtagonsand hexagonsl dothneatlydeclarc how and nature Geometrizeth obserzteth in all order things. -Sir Thomas Browne




t or soworkthehoneY' bees, that Creatures bYa rulein teach nature to Theactoforder a peopledkingdom'

Honeu Spoce is architecture built by bees' The most famoushexagonal Eglptianstook thehoneyh";;ts. The"ancient ;;;; ;ffb ; th" supremes)'rnbotof structure-function-order only tutoredbrsects' i t fissonsto their society' build ".J"ooU"a principlesthev know by instinct' ;";5;;h",""a1

strength' J"%-u"t *a maximize interior spaceand Honevbees,one of the very few domesticated,msecrs -William ShakesPeare (silkworms are another),build their honeycombm altematbeesnave irs hexasonal tiers like a wax crystal Atter the wax oozes e#n Ur:g" qugntities of -the sweet substance' this.wa1 from speiial'ptckets in their bodies' Thgy'rhew patuntit iiis soff and work it into the familiar hexagonat strength of the hexagonal the space-fiiling il-;; ;; lomts/ tne structure of three-comer 120-degree shape and to hold the most honey: a mete one and i"*i *"- it "t"a one-half ouncesof wax holds four pounds of honey' Homets and wasps,on the other hand' alsobuild hexago.ui ,,"rt, but of mud or a paperlike substance-whose strengthalso relies on geometry But unlike the beeJ honey T herebeing,then, three combi, which have a horizontal entrance, wasps' nests hang utta are entered from beiow, while hornets prefer figures which of themseloes ""tti.utty their hives from above. to enter arounda can up space fill


joints *"air-'i't'*i*' ti'ree-comer tomhi-

point,aiz . . . thetriangle' and square thehexagon, wiselY haoe thebees selected their structure for the that whichcontains susPecting moslangles, thnt it couldhold indeed morehoneythan the other two' -Pappus (c. 300.t.o., Greek mathematician in Alexandria, Eg1pt, whose writings stimulated seventh-century geomefric studies)

of Different views into the structure-function-order a beehive'A nestenteredfrom below. wasD's



FRAMEWORK DODEKAD_TWSLVE_THE NUMBER Twelve, called Dodekad (in Greek literally "two ten")' not from the principles of the Hexad' Although .*.t*.t Pythagorean wi*rin the firsi ten terms of the i""ffi"a "virlutui, it t"tut"t intimately wiih eachof thern, even the (seechapter seven;' gin" '"- number seven e"- io"uf" six, ihe Dodekad extends and refines the Hexad's structure-function-order comprising whole events' The wonder of twelve is that it has so many divisors (one' two, three, four, and six), making it the supreme m-rmber appointed at the archetypallevel of mathematicsas the nat,riil ftu-e*ork of arithmetic and geometry' Its ProPerties have made it the number of symbolic measurement and sacred structure since prehistory. Construct the Dodekogon Among the many met}ods for constructing a tweJvgoointei or twelvelsided structure, three stand out for their easeand antiquity. They are based on the fact that twelve is divided by th"e", forrt, and six, the only three numbers whose shapes (triangle, square, hexagon) tesseliate, covering a flat swface without leaving gaps. Use your tools to try eaih of these methods. Be free to explore each construction further.

First method: (1) Construct a hexagram star within a circle. (2) Draw straight lines across the circle's center through each of the six crossings, extending them to reach the circle. (3-4) With the original six points we now have twelve equally spaced Points around the circle, like a clock face.Discover the star's pattems by doodling, connecting various points with straight lines or sewn thread. Use the new crossings as centers to fum more circles.








before Geometryexisted thecreation. - . -JoharnnesKeprer Secondmethod: Construct a square and divide its sides into four equal parts, drawing a grid as shown. Inscribe a circle within the square, a Monad within a Tetrad. The circle crossesthe grid at exactly twelve equally spacedpoints, like the numbers on a clock. Twelve-part wholeness emergeswhen the transcendental Monad reconciles with the rational Tetrad. Twelve represents the crystallized expression of divinity.

Third method: Construct a square and inscribe a circle within it. Keeping the compassopen to the circle's radius, place its metal point at each of the four comers and midpoint of eachside. Make an arc (or tum the full circles)from eachpoint. The arcs cross the original circle at twelve equispaced points.



Twetveond Thirteen Twelve is related to thirteen, the notorious but misundernumstood number of superstition lA4riletwelve is a solar of a solar year flt, ,nitt.." is lunar, as the twelve months t}nrteen lunations' The relation is more obvi"o"tui.t ""utly look geometrically Each of the methods for ;;; ;;"" *6 resultsin twelve equally sPaceo a constructing dodecagon TWelve points ooir",tt uto.rila a thirteinth one at the center' of the crossings a hexagramstar around the thiri.*i"t as rt teenth at their cenier. We'll look at this geometry through the cenevolved in many symbolic expressions turies. The pattem of twelve and thirteen continues in three dimensions.Exactly twelve equal spherespack preciselyto enclose a central thirteenth sphere' If vou have some clay and toothpicks, make thirteen balls and ioin them to seehow twelve fit around eoual-'size or'"kirr" one at the center.Each spheretouchesthe center and four others around it in an omnidirectional closestpacking. If vou remove the central sphere, the remaining iwelve" collapse siightly to create an icosahedron, the twenty-sided Platonic Volume. Known and studied in ancient times, this volume appearsnaturally in galena and other crystals' Crystallogrirrhers kno* the shape as a "cuboctahedron" You can -lk" orr" from a cube of clay by slicing tetrahedra off its eight corners,from the midpoints of its edges.



Cuboctahedron crystal is a $aph of balanced forces, or "vector equilibrium," in twelve directions out from the center.

The cuboctahedton (or "dymaxion" as it was called by Buckminster Fuller) is an omnidirectional graph of forces in balance. Its twelve lines radiate from the centet as a "vector equilibrium," or truss, balancing t}ree-dimmsional forces most effectively. Anv wav we look at it, in two or three dimensions, twelviaround-one comPoses a wholeness, unity., perfection, a universal module of structure-function-order'

ZODIACAL SOCIETIES one A fatherhastwelve children.Eadr hasthirty daughters, side white, the other sideblack,and though immortal they all die. who is thefather? -clrobutu, (Answer:the year) The pattems by which the cosmic creatin! process construcls the universe are consistent on all levbls. Wherever efficient structure-ftrnction-order is required the Hexad aPPeals. The prehistoric transformation from nomadic hibes to settled agricultural societiesoccurred at about the sametime around the world, soon after the ice age ended about 12,000 B.c.People brought with them their myths and understandings, which included tales of the sun and night sky, as wel as a surprising knowledge of mathematics. The daily cycle of the rising and setting constellations that had served as the



tluough the yeal indicated nomad's clock as it progressed periodsof soil prepiration,planting,care,and hgrvegting calof the determines structure the agriculfural and therebv endar. Contrastedwith our modem civilization with its techwereprimitive But prehistoric societies nolosicalwonders, thinkingwasmuchmoresophisthat evidence their there"is ticatedthan we might imagine' They often consciouslymodon eled their commr-rnities the twelvefold pattern of constel' lations they perceived in the night sky. To the tribe, the whole natibn s)rynbolized the twelve-constellated zodiacaL wheel that they replicated on the landscape, often reshaping it with mounds and ditches. They divided themselves into twelve provinces, each corresponding to a zodiacal morrth' Coins unearthed by archaeologists often show the zodiacas)'rnbolism of the city at which they are found- !1ch province was divided into thirty clans like the days of the month, which were further subdivided into thirty houses. The Greek leader Solon devised his famous Athenian constitution this way. Each province was also allotted onetwelfth of the national myth, and each choir sang music based on one note of the twelve-note chromatic musical scale. To the mathematical philosophers of ancient Greece, "heaven" beyond, which encompassed the four elements, was svmbolized bv the fifth Platonic volume, the twelve-

The twelve-part division of ancient Ireland with its center at Tar4 the High King's celestial court, was designed as a microcosm of the zodiac.



this form' Plutarch,in discussing. faced dodecahedron.

At the same ifrl.rn-.f"?tt *iahin eachof twelveprovinces and thecycle ir*Jt r"r*t""as thefull circleof 360degrees teshvalo.ays)' . of the veir of 360days(plusfive intercalary * 'Tl-,f";;;;i u"i"rii of this division of a nation is that could t*utu"'t many divisors'the society *lafl ,i."tt"-i"i trut' be easilvsubdividedto perform differenttunctrons' by that env thephilosoptrer-leaders isioned ;;il";ffi, tr they wereparticipating the night sky on earth ;;;; the - '' harmonies o{ the cosmic structure-iunctron-oroer' book at i.n ftff.ftel and ChristineRhoneshowin their informaNations,thete is abundant historical The kgistator's iobis f*iitlniU, first tf,at the ancientsgavegreatimportTtu. t: u iion to the to locate citYas "onfirm of society'Many of these"twelve-trlbe order in as precisely Possible the twelvefold cities'myihs,or by ;* revealed thiir tweive-gated. ;;;; thecountry. ' . of cosmoloSlca v center twelve disciples around a central figure' and Nexthemustdir:idethe ordered roYal courts. The populace of these twelve-tribe nations participated into twelae country of pilgrimages, festivals'and a circle of But sections. first heought - *^-i"ii.o"tse terrestrial oerpehral choirs following the ieasonal flow of area a to rcsefi)e sacred for of their landscapeAtthe zodiacil wheel i".iiri," a Hestia,ZeusandAthene *hen ;;, "t ""J,he t1'teearth's magnetic energy peaked' ill sprtlp/ (callingit the " atoPolis"), maior festivalwas held at the holy site-temPle' holy approPdate lor that monm' its and enclose boundaries; weil, or cavern-in the province pt useof tne annuil festival iound was choreographed hewill thendiaidethecitY naci )in ,unu" *ith that month's zodiacal constellation and to b" thewholecountrYinto i""ttt"J ty-U"fs and characteristics of that sign Animals of and bY sections lines twelae zodiacal ,ignifiautta" were consecratedto the appropriate when Taurus rose with the sun' a bull radiating from this central "iw. On.irritft" t"onth Aries rose, a ram was consecrated point . . . He shoulddioide *j, oto-il"^t; when Pisces,a fish. into tweloe and, during thepopulation G"" a'year at midsummer sunrise, the whole populas e c t i o n s , . . . [ a n d ] . . . tion would convene at the center, the thirteenth site' for a diaidethecitYinto twelzte e;at festival. The national choir, comprising the complete usical scale,would lead song, dance,and harmonization waYas in sections thesame of the whole. Ofterg as in China, Britain, and the Americas' theydioidedtherestof the and mounds to


face pentagonal intotT't oi*9111::?"Te^ Jiula"t tne rePresents "u.nin all. Thus,the dodecahedron ?60triansles


. country. .

to Now that we'oedecided into diaidethecitizens

the landscape was reshaped with ditches enhance the flow of terristrial energies' Attuned to each other and to the subtleenergiesfertilizing the land' the choir would amplifv the natural-fusion of terrestrial forces with


203 A clerk, astronomer, and rccorder (Psaltu ofBlancheof Castille)shtdY celestialgeometry to calculate festival dates' The picture itself was composed with the same twelvefold slnnmetrY the three men study in the heavens.

celestial energiesand literally enchant the land and the society. In this way the society was ruled through music. Although ancient knowledge of sound vibrations is a longlost science, it is slowly being rediscovered tfuough such practices as voice toning and experiments in c1'rnatics(the process t}rough which sound vibrations establish geometric pattems in matter). Countless civilizations are knonm to have regulated themselvesby the twelvefold pattern. According to the early Greek historian Shabo, Egyptian society was originally divided into twelve noma, or provinces, each ruled by a king, thus extemalizing the myth of Osiris and his twelve retainers. Greek legend and myth are full of reference to the duodecimal pattern. Foremost was the Olympian pantheon

u)e sections, should twelae (after it's all, try to realize the clearenough) of fiumber enormous of the dioisors subdiztisions and haoe, section each in howthese turn reflect and canbesubdi7rided again. . . This subdiz:ided is themathemntical whichwill yield framework youyour brotherhoods, ativ localadministr e units, your military oillages, anies mar gand chin comp as columns, wellasunits and of coinage,liquid dry and measures, taeights. all Thelawmustregulate so these details that the and proper proportions arc correspondences obseroed.




overby Presided -1"1:,t:.:l: of twelvegodsand goddesses wrtn wascrowneo no""ri",creei sungod (sunking)' "".i"t. odvsseui saiiedwith twelve;;i;;';'";;;'' twelve phasesof any wrrote' The legendary Cadmus is

ffi ;"fi; il;'thehu ar #"":i3;?ffiT:#:1 .il" : :*:i'; HerculesexPress nature saro ill;;"';;;;;i--t(" bv .""t"' of rhebes drivinga stake

il;;;; ;ili;;;;


;;'"li;' :::'J"ri:HiiiJt::-::t il;H;transition ;iI: luna TH from see the natio-nal nto of its twelvefold society' the sacred "world l'ua a stone omphatos'-or c'"e ;;;i;;-;-u"tpr'i Delphic o'acle sat on a golden tripod Ti';;;;*h ;;J" verse later ir-ru aurr"* orra, a fissure and spoke in hexameter -* used bv Longfellow and other poets' foundation myth twelve vultures.appear n*if." n"?* hllls) tor tne to Romulus indicating the exact site (on seven Latin word for citv. Refefiing to the terresrrial zodiac' the ;-;i;;1;rbtl is relat"d to the word "round" (orbrs)'Roman bronze r"*' *It ii"pr"yed in public written on twelve tablets. twelveRelisions and govemments mirroring archetyPal soclness ab"oundedthloughout the world's early settlecl. the W"'rr" heard a6out the twelve tribes of Israel' nobles (aswas the Norse "tiur. fi"gt t"ttounded by twelve V"ai" odi^" by his twelve counselors), the Dalai Lama's kish High ";J ft"La C."*ff of twelve great Namshans' the tne Court at Tara, and Charlernagne'smystical 9o]llt' -.rn Arthur and his Rourrd epic legend of King ;;t""-fart the .r t*"rti r..igf,tt, Kittg n'ttttt'trrepresents sun and -".".rna i"ui" table/is, of c-ourse,the twelve-constellated tn" zodiacalbelt. Native America was once a twelve-tribe nation' eacrr tribe consisting of twelve clans' The traditional sweat lodge Native American tePee1s The and tepee stnictures are still ritually constructed to resemstructured as a calmritually pattem of a calendar resting on twelve-poles iiu tL"'-"t"i" dar. of the months, with certain obiects and symbols of each month appropriately placed around *re periphery' Less'well'known is the existenceof the twelve-aroundtt"til"i*

tne became Sowingits twelve"teeth"' which

e.a11 ure.tgrlitizins harnessing



one pattem in the govemment of the United States, the founders oi which wire Freemasons.Seeking to establish a utopian society in the ancient ffadition, they gave the go't'ernment a classic twelvelold structure-function-order. In every way possible they incorporated zodiacal s)rmbolism' George Washington kept a circle of twelve generals aror;nd him to fight for the thirteen colonies. The s)rmbolism of thirteen stars and thirteen stdPes on their flag, thirteen lettels in their motto, "E Pluribus Unum" ("From many, One"), and thirteen buttons on the sailor's uniform accented their intent. The number of colonies was not accidental but a hidden re{erenceto an ancient teaching. The country's founders put the two-sided Great Seal of the United States on every dollar bill. In chapter three we saw the seal's Egyptian pyramid with thirteen courses of stone capped with the triangular eye of divinify. The other side of the seal depicts the great eagle as the Egyptian hawk Horus. Above the eagle's head is a glory of light and clouds enclosing pentagonal stars exptcitly displaying the twelvefold canon. When the glory's geometry is magnified and applied to the seal as a whole, the underlying plan on which plain. it was composedbecomes

DISCOVER CLASSICTIEXAGONAL IN COMPOSITION ART Try this with tracing paper over the Great Eagle Seal on a dollar bill: out a Doint at the center of the circular seal at the eagle's cheit jusfabove the shield. Put the point of the compass there, and open it to the circle's radius. Turn a circle. Confirm that it matches the seal's circle. Starting with the compass point at the top, inscribe a hexagon within the circ1e walking the compass around its circumference. Conby nect the six points to inscribe a hexagram star. If your pencil is sharp, inscribe hexagrarn stars within eachinner hexagon. The construction on the tracing paper will reveal the obviously planned composition of the seal underneath as shown in the illustration. The identical hexagonal skucture underiies a large silver plate called Daaid Trying on Saul's Armor from a similarly



Eagleside of the U.S. Great Seal

Armor Daaid Trying Saul's on

Aegeus before the Oracle of Delphi



Bella Coola wooden sun mask

British Royal SeaI

designed set of four plates from Byzantine Constantinople (610-630).Although it was designed ovet one thousand it years earlier than the U.S. Great Sea1, employs the same underlying framework {or the proportionate placement of figures. Note how the " omega" near the top corresponds with the glory of clouds above the eagle. S1'rnbolsand art of many societies that centered around a sun king were often designed upon this hexagonal laftice, a symbol of divine order on earth, the pattern of solar societies since prehistory. The twelve dark squares around the painting on a plate from Delphi, a twelve-structured sociefy, gives us a hint as to the geometry r.rnderlying its design. From Native American sun masks to British Royal Seals, hexagonal and twelve-structured geometry symbolizes both the form of society and the perceived cosmic structure with which it is meant to harmonize. TWo hexagons (twelve points and crossings)have been marked around the Delphic painting and Native American sun mask for you to connect in order to discover their geometric guidelines.






th".p!lnature In their desireto live in harmony with "ld heavens'the-"":i:"1" IJ"a rtrr"*r"-function-order of the their to everydetail.of cosmit ;;1t"fiil;;1.'"fold "a"on of societv Not onlvwastheoverallpattem ;J.l;;;i;i;;;. of but frame, multiPles six'(par#ililil;;uodec#al to coordinate


(distance' area' volume)"of measuresof structure and sPace ^foi-r",ior, and of order 1*.igtt s, coinage, administration) (timekeeping). 'its workings' -Cr""t, ff,e iur,J., even integrated language into Syrllke Egyftian, Hebiew' Arabic' Coptic' er,.i"nt


were zoo,600) used eo,

il.Jrj ;;;;;;'

words of mvth and cosmology revealed their significance'

used s;ktt, (iteilfly "we11-constructed") letters its lettels'important tr uaaiogtl'u valuesof

* io' tl'eEgvpfr riverNile'NEIAO!'transTh;e;#;" to refers the ;;;;;;;0;' rheir sum'365'

We've inherited our "English system of measure" from the ancient twelvefold canon, Although the modern metric system derives from the aftermath of the French Revolution as a measure of the earth (the meter is 1/10-millionth the distance from the North Pole to the equaior ihrough Paris), it is not Part of a comprehensive system integating celestial dimmsions and cYcleswith human iife.

Structure (Space) 12inches=1foot 36inches=1yard 6 feet = L fathom (from the Saxonfaetm flot "embrace" of extended arms) 660feet = 1 furlong ("furrow length," the distance.. stretchedout in a straight line) plowed in one-acre, = measure of a circle 360 decrees 60 minirtes = 1 degree of arc 60 seconds = 1 minute of arc Function (weights and other measures) L2 = 1 dozen 12dozen=lgross (originallY) 12 ormces = 1 Pound Order (Time)

60seconds=lminute 60minutes=1hour 24hours=1day

30days=lmonth 12months=1year 360(+5)daYs= 1 Year



Nile's 365-daycycleof rising, overflowing, depositingfertile silt, and receding. The Egyptian number word for sixty was l'h"r,." the Greu"k word'-heiad("unity") has a letter value ol sixtv. And the Greek word for universal strucfure-functionhas a value of 600,emphasizing ordlr, KOIMOI (cosmos), the importance of six and its multiples in cosmic design' Ttie modem world has inherited remrlants of the ancieni system of measure from Chaldea through Babylory where I o* noton whowas tire captive Jews learned it and passed it on to Rome-and of bornin thepossession it Europe.In the United States is known asthe "English" sysI am whois knowledge; one tem. and Over the centuries, the ancient relationships have fondofantiquitY, changed somewhat. The pound weight was originally 9qual it in earnest seeking there. to twilve "ounces" (from the Latin uncia signifying "one--Confu cius (551-429 twefth") and only later became sixteen ounces.From the B'c., the British derived their monetary court of Charlemagne Chinese phiiosoPher) system o{ twelve pence to the shilling and twelve- shillings to the pound sterling, used until the adoption of the metric '1,971. Tlne twelve-person iury derives from this system in canon, as does the analog clock face. The metric system makes calculations easy but dissociatesus from a frame linked with the cosmos. The systemof duodecimalmeasureis so widespreadone would think various civilizations had collaborated on it. But independent of each other, China, Babylon, Egypt, Greece, and India all divided the daylnlght cycle into twelve twohour periods (which the Greekscalled "Babylonian hours") and sixty twenty-four-minute parts, further subdivided into sixty 24-secondparts. A Chinese clepsydra (water clock) dripwas made of six water pots in descendingsequence, ping from one to the next. A Hindu hemispherical copper bowl called a kapalahad a hole in the bottom; when floated in a basin of water, its hole allowed in just enough water to slowly fill and sink it sixty times in twenty-four hours' In ancient cultures timekeeping meant more than calculating festival dates and hours for Prayer. It was a way that This ancient Chinese clock, hurnans could seethemselves integrated within the cosmic like a mandala, was also a scheme.They accomplished this by making the basic unit of candle. As ii bumed along its measure equal to the duration of one human breath (= 4 sec- labyrinthine path, it marked off the minutes and hours of onds). Structure-function-order were united and measured the day. in terms of the human breath as distance-breath-duration.



found their place of by Enlareine scales six and sixty they and stars'. u*o.ti ti'u iimensionsof the earth,sun'moon' showsnow trus ancientHindu systemof measure Th-e works. system 1 prana (breath) 6 prana 60 prana = 4 seconds = one vinadi = 24 seconds = 10 vinadi = 240 seconds = 4 minfor utes = time necessarY earth to on its axis rotate 1 degree = one nadi = 360breaths = 1440seconds = 24 minutes = time necessary for earth to rotate 10 degrees = one sidereal d ay / il},|tt = 21' ,600 = 86,400 seconds= 1440 breaths minutes = 24 hours = time necessary for earth to rotate 360 degrees

60 vinadi

60 nadi

and thejr The people of ancientcultures saw themselves and stars' Obrrro, duemeasure, for .rrlt rr" dr"uihit g in unison with the turning earth The numbers reveal the cosmic structure they Percelveo' righttimingis in all = day (86,a00 12 x 12 x 660)is imPortant The number of secondsin one of the sun (864,000 12 x12 the things most = exactlv one-tenththe diameter factor. in x 6,00b) miles.The numericalvaluesof the Greekletters oi ih" ,ru^" of Pythagoras, named after the Pythian oracle -Hesiod (c. 8000.c.,Greek adds up to 864'a cerwho predicted hi'sbirth and greatness, Poet) tain referenceto his solar stature' (= And the day's 21,600 6 x 60 x 60)breathsis exactlyten times the moon's diameter of 2,160miles' Astronomersalso mo* tnut the sun travels at 21,600miles per hour through the ealaxv as a satellite around its unknown center' This t}:.e ,rrr#bu. ii also one tenth of the mean distance between miles = 60 x 60 x 60)' which is 100 earth and moorr (= 216'000 times its own radius. The moon's diameter in miles resonates with the 2,160-yearmeasure of each Platonic Month' or Zodiacal Age. Tweive such agescomplete a Great Year' of one cycle of the Precession the Equinoxes,as the eartn n 25,920(= 6 x 12 x 360) years' slowly wobbtes on its axis fne rinlt of tlnemile, which allows this hidden structure of the cosmos to emerge, is much older than its attribution to of 1,000(in Latin mllle)-paces the RomanLegion impli:t T" mile was defined so-that about 25,920of them equal the cir-



culnference of the earth around its equator, in resonance with the time of a Precession. There's little sensein arguing whether the ancients knew the dimensions and cycles of the earth, moon, sun, and stars' We know so little about the seersof old and the deep antiquity from which theseunits of measure derived that our concluiions, colored by today's very different vision of our place in the cosmos,can only be speculative.But by noting ihat the number values of the Greek words for " year" (etos = 575) and "brearh" Qtneuma= 576) are just a single unit apart, we can get an indication of the basis of ancientmea' sure. The consistent predominance of multiples of the numand 500indicate that the manifest cosmos bers 6,12, 60,360, is an approximation of its mathematical ideal. We can marvel at ilie ancient systems of measure, constructed as they were with a sophisticated knowledge of the archetypaframework numbers and incorporating everything from a single human breath to the size of the sun and +he 25,920of year cycle of the Precession the Equinoxes. are number correspondences not coincidentalbut These perceived as an integrated recur wherever the cosmos is whole. The ancientduodecimal units of measurewere carefuJly chosen and coordinated by seers to integrate human structure-function-order as distance-breath-duration with celestial rhythms and dimensions. V/hen appropriate units are chosenthe celestialpattem shows through. While we measure with rulers and weigh ourselves on most peopie don't realizethe vast and intelligent sysscales, tem we've inherited, mirroring the cosmoswithin which we live and function. These tools are fossils of a long-forgotten way of looking at ourselves as integrated within universal relationships.Ttris knowledge of ourselvesas part of a harmonious whole is now dimmed to everyday awareness,but it can be part of a great vision to be reclaimed.

whichhas A tradition beencreditedby many Iearned menoaerthe is centuries that the their ancients encoded knowledge the laorld in of thedimensions lheir of Tnonuments. sacred -John Michell

TEMPLE THE RECONCILING Strict morethanjusi style. design conveyed In ancient tirnes rules based on archetypal symbolism existed for the design of everything from kitchen utensils and fumiture to temples and



-{i moflrnents' Templesand societywere ot PeoPte' ate and harmonize the heavens with the affahs was Through number and geometricsymbolism cosmology ot sacred.arcrutecbuilt i-ito the measures and proportions tempre ture. From prehistoric megaliths to Gothic cathedrals'

witli profoundand surplsinq kno-ryIa"ri* i. of "it'tU"aded t"futio*t ip, u-ottg th" d'i*"*ions andrycles the "Jniof-.o", sun, planets, ittd ttutt' ln accordancewith the ""iUt twelvefold structure, the temple was designed as a overall caltimekeepingdevice,a measuringtool, both a clock and a heaven on earth' endar mirro'ting ttre pattems of "of h"un"t, and earth was symbolized , in Harmony ti*"'t by a geometric puzzle posed-by the,Delphic ut "i*t Oracle and known ls "squaring the circle " The challenge a ,qlrute *hoteirea or perimeter was equal *ut io "o*tto"tcircumference of a given circle' This puzzle bas to the area or scoresof people over the centurieswithout soluobsessed tion because it cannot be solved with the three tools of the geometet requiresan impgssibilt!: ihe absolute reconciliationbetween the transcendentalcrcle, whose immeasurable circumference teases us with a never-ending decimal, and the square's,rational sides' The ideal and tG actual, the archetype and its representation' can never be identical. A "squared circle" is the ultimate symbol of reconciliaheavens with the earth. Thus, it is approximated tion of th^e in the srrmbolic geometry and measures regulating the architechre of maiv tempies, whose role is such a reconcillaflon.

Squared circle. A square ard circle of nearlY equal Perirneter and circumjerence. Perfect equatty is impossible to _ achieve with the geomter's tools.

Although a squared circle cannot be constructed precisely with the giometer's tools, fwo aesicaspisces constmited at right angles to each other yield a good approximation, which was known in ancient times. The! square formed by the four irurermost crossings has a perimeter nearly equal to the circum{erence of a circle made within the pisces.T\is symbol of geomefric harmony becamethe tsesicas core of the temPle's design.

A Byzantinecupolareconcilesthe "heavenly" sPhere with the "earthlY" cubq forming a three-dimensional versionof the squaredcircle.

Squorethe Circte a creating a fol construction squaring circle, Do thisancient of the approximates perimeter a whose circumference circle



square. Ordinarily, it would be done ritually on the earth at the site of the proposed temPle. pisces right anSlesto at 1. Construct tw o uesicas each other. 2. Corurect their four innermost crossings to make a square. pisces 3. lnscribe a circle within the two oesicas through the square. This is the "squared circle" whose circumference approximately equals the square's perimeter. Confirm it by measuiing each figure with a piece of string. 4. lnscribe a circlewithin the square,representing the earth.

Mrorrrrr, to plotthe ,. sley . until at I use ruler,thus, a length Thecirclehasbern and squared, in themidst place set, is from A market whichthestreets as Are drawn,to radiate ftom a star, itselfa Thebeams uthich, of circle, shine . Straight forth to eaery point. -Aristophanes (c. 450-{. 388 n.c.,Greekplaywright)



5. Atop the square construct a small squarewhose centeris on the outer ring. Inscribe a circle within it, representingthe moon. . The geometry of the squared circle discloses the inner harmony of the earth-moon relationship. The inner circle of earth within the square and small circle atop it (the moon) show the relative diametersof the earlh(7,920=8x9x10x miles).On eachside of the moon's 11miles) and moon (2,160 square is a three-four-five right triangle, easily formed with the use of a twelve-knotted cord. In this way the ancient seers studied the archetypal Pattems of geometry and enioded celestial dimensions within temple design. The relaMusic was also incorporated into this scheme. tionship between the diameters of the large circle of the moon's oath and the circle of the earth come closeto the ratio of nine tb eight, the basic interval of the "tone" on which the musical scaleis built (seechapter seven).The ratio of the areas of the circles is close to @,the golden ratio of natural growth. The dynamic cosmos was worshipped as visible, living rnusic, its celestial dimensions revealing an expanding musical scale,the Pythagorean "music of the spheres." 6. \ /hen twelve such "moon circles" are constructed within the band around the "circle of the earth" as showrL and a hexagram star is con-



structed within the band, we seea more complete scheme rich in number relationships. For example, the combined radii of the earth and moon equal 5 , 0 4( = ] . x 2 x 3 x 4 x 5 x 6 x i = 7 x 8x 9 x 1 0 ) 0 miles, the number of the "ideal population" of Plato's utopian twelve-tribe nation of Magnesia Plato disguised thesenumdescribedin his Lazus. bers of the cosmic ideal throughout his writings to hint at the numerical and celestial schemehe wasn't allowed to discuss overtly. The celestial design canon contained in the Delphic riddle of the squared circle served as the pattem for twelve-part temples, rnyths, measures,and societies throughout the world over many centuries. Stonehenge and Gothic cathedrals are examples from the long tradition of temple design that encode such knowledge in their geometric Proportions. Discover the Geometrg of Stonehenge ond Chortres A squared circle has been constructed on these images of Stonehengeand the altar at Charhes. Explore their geometric scheme using your compass open to the radius of the inner circle. Put the point of the compass at the top of that

The North RoseWindow of the Cathedral of Notre Dame at Charhes displays the twelve-part structure of the ancient caron.

Stonehenge constructed was fust of wood around2700 8.c., later of sione, built in stages endingaround1500 r.c.



Stonesplaced along Stonehenge's outer circle, rePresenting the moon's Path, were used to predict the moon's risings, settings, and eclipses, as well as solar rhlthms. The altars of Chartres and other cathedrals were designed wiih the squared circle and hexagonal fusion geometry of Stonehenge, as seen in its Rose Window, which repeats the twelve-part geometry of its altar.

circle and walk it around to mark the six points of a hexaeon. Connect the points to draw a hexagramstar within the i"r.ugon, u, ,"".t earlier in the geometry of the GreatSealof the tinited States.The geometry of Stonehenge'sstones and Chartres' altar area are aligned with these interlaced kiangles. Symbolically and functionally both are temples that ierved to fuse and harmonize the energies of heaven ancearth, sky and soi1,the entire cosmos and a single human breath.

SELF THEsUN-BORN Modern scienceknows that there are twelve chernical steps, like twelve notes of the chromatic music scale,in every spiraling turn of the DNA helix, determining our genetic makeup. The ancient mathematical philosophers, not having the tools of modem sciencebut seeing how the number twelve relates to all numbers, recognized that the functions of all numbers group themselves around twelve. With the unchanging properties of number as a modef they codified this as a natural frame of the universal structure-firnctionorder. While the twelvefold cosmic canon served as model for all aspectsof society, there was another level to its sym-

the Donot seektofoltow of footsteps themenof old; u)hattheysought. seek -Matsuo Basho (1.6&-"1694, Japanesepoet)



bolic architechue,myths, and measures A personal inner significance was given to initiates who had achieved a de"greeof psychol'ogicalpurification and were ready for m&e seriols gto-th und spirituai transformation' Since thev were stricilv confined to an oral traditiory the details of what they wereiaught is not known. But it's no secretthat many of the mystery schools of ancient cultures sought a spiritual rebirth within the individual using the zodiac as a teaching tool in an original twelve-step self-help program' Froir myths, legends, and art we know that the zodiac was used as a sublirne allegory of initiation. The twelve cryptic zodiacal constellations were seennightly by everyone; for the initiate, they were translated into tools of self-knowledge The zodiac described the inner constitution of humans, the structwe-function-order of ourselves, mirroring that of the universe. According to the ancients, the archetypal principles revealedby arithmetic and geometry, rePresentedby the iodiac, seen rnanifest in the designs of nature, technology, art, the clock and calendar, temple architecture, myths, mea-

read slory W oildst the Kng? selt'-born ofthe the Firstlearn sPlendid ofthesun, language the of Thespeech thestars, coy ffioon's whispering, of Themusic thePlanets, andof one, Earth, OurMother her cro oning cradle-song babes, Toheruncounted gain who,whentheY to Thesoul's stature, fuII . belong. . thehwaens -James Morgan Pryse

Mithras, personification of the Higher Self, is represented as the archetype of the year. Surrounded by the zodiac, his spiritual powers, he pips his energy field and bursts from his Orphic Egg, bom into identity with his Higher Self. The Greek letters of the name of Mithras, Mer,0pao,add up to 365, reinforcing his symbolism as the Monad of the year.



The Circle o{ the Zodiac as the Quest for Self. Fi-fteenthcentury personification and zootyping of the drama of the energies and principles of the sky by Barthelemy l'Anglais (Book theProperties on of Things).

within each havecorrespondence sures,and whole societies of us. The initiate was taught to seethe world inside-out,to twelve-part wholesasoarreflection. see This is the inner significanceof the dodecahedron,the fifth Platonic volume with twelve five-sided faces, s).'mbol of heaven.Five is a symbol of humanness.Each face represents a different matrix of ordeals and opporturdties before us. The whole dodecahedron represents "heavery" our spiritual completion. NORTH-Midnight-Winter soL stice-Darkness. Sun risesfurthest south, bavelslowestin the sky. I SeekMy Higher Self Through What I Use-Capricorn .m :,

I SeekMy Higher Self Through Hurnanity-Aquadus

I SeekMy Higher Self ThereforeI Arn-Sagittarius

I Seek HigherSelfand I My Don't Seek HigherSelfMy Pisces tr-c R I Seek My Higher SelfAries

I SeekMy Higher Self Through What I DesireScorpio I SeekMy Higher Self Through What I Balance-Libra I SeekMy Higher SeiI Through What I LeamVirgo

# > rJ: ir o-E:. n F E

q 1 4

tt! g d , i L- YZ I Seek My Higher Self Thrcugh What I HaveTaurus

x ",: E.

I Seek My Higher Self Through What I Think-Cemini

I SeekMy Higher Self Through What I Create-Leo

I SeekMy Higher Self Through What I Feel--Cancer

SOUTH-Noon-Summer solstice--4reatest light. Sun rises futthest north, soars highest in the sky.


Although different myths were conceived for different cultures, eaih in its own way depicted exploits of the unified Cosmic Being,the ideal of every age.The twelve attributes -by the zodiac were assembledinto an ideal represented teicher, who guided the spiritual welfare of humanity durperiod. In their cryptic language,solar ine that 2,160--year desiribed a plan of spiritual birth within ourall"egories seGs, of purification through twelve tyPesof ordeal' The twelve signs of the zodiac, twelve disciples, twelve tribes, and pantheon of twelve gods and goddesses represent twelve archetypal principles within everyone' These are our own twelve phases of transformation, the soul's ioumey home with Odysseus through twelve ordeals, the itages-of expanding growth necessaryin order to become alJ coriplete atta erlifi ienea. Each individual comprisesaspects of the whole, and we will eventually find twefve within ouiselves the principles represented by the syrnbols. The twelve-part national myth of twelve-tribe nations_is a tale about both ttre visible sun and the spilitual light within ourselves. A savior is bom during the darkest days of the winter solstice; his light grows and dims through the twelve signs of the zodiac. As the srin goes through the hours of the day and months of the year, we experience a correlatedprocessof development.This processcanbe seen in a great deal of Egyptian art, which is often composed on the grid of a calendar. The myths and legends of Apollo, Heracles, Osiris, Ra, Gilgamesh, Odysseus, Jesus, Mohammed, Arthur, the twelve harvesters in the Egyptian Field of Amenti, twelve carpenters,potters, weavers, fishermen, twelve rowers in the ship with Ra, twelve stones chosen by Joshua, and twelve facets of the ancient solar deities all s).rnbolize our innate faculties of spiritual intelligence, twelve grades of being, twelve stagesof un{oldment. These twelve phases of transformation are twelve ways we seek identity with our Higher Self. This drama is repeated on all levels from the subatomic to the supergalactic as the great and mysterious power of consciousnessin the universe develops through a round of fwelve stagesof identity ai eachlevel. The ancient Chinese system of acupuncture recognizes our body's twelve-part s}'mmetry as having tweive "meridians," or subtle nerve channels, around a central thirteenth

in Astrology is a science an and itself contains of body illuminating me taught knowledge.It manythings,andI am to indebted it. greatly eoidence Geophysical of the reoeals power the in and starc theplanets to relation thetenestrial. In turn, astrologY to this rcinforces power This some extent. is whY astroloryis likea lifegiaingelixirfor mankind. -Albert Einstein

you, soI drew I looed thesetidesof men into my and wrotemy will hands across sw in stars. the -T. E. Lawrence("of British Arabia," 1888-1935, archaeologis! soidier, and writer)




monks, in And when, these four nobletruths my and dueknozoledge insight sections and with thethree waswell twelz.te diaisions . purtfied,thenmonks . .I hadattainedthehighest complete enlightenment. -D11ctOna


This Chinese acupuncture map of the body, a replica of the cosmos,recognizes twelve subtle nerve chanaels around a central thirteenth, an archetlpe of the zodiac and twelve spheres surrounding one.

meridian through whidn ch'i, or creative energy, flows. This is the sarne fertilizing ch'i thal courses seasonally through the body of the earth and that twelve-tribe societies sought to amplify through their temples and festivals. Although dowsers are known worldwide, the only ancient fertilizing system remaining intact is the Chinese scienceof feng-shui (pronounced "foong-shway," literaliy "wind and water"), the geomancy of orienting and siting temples, tombs, homes, and societies {or the purpose of "enchanting" and fertilizing the 1and, family, and nat'ton. Feng-shulbrought harmony to the envilonment in ancient China as acuPuncture harmonizes the energies of the body. The initiate was instructed to know him- or herself as a dymamic energy system like that of the earth. Consciously directing energiesto flow in constructive channelswas done for the human's body and the earth's.Beforeone could reach enlightened awarenessof his or her identity with the whole cosmiccreatingprocess, was necessary tame and purify it to both the energies of the land and the initiate's turbulent inner "private world." kr an advanced stage of meditation the head's twelve craniai nerves are energized to iiberate the individual as an enlightened being/ a soiar conqueror, a "self-rolling wheel," world savior, avatar, messiah of the age,like a central sphere surrounded by twelve others, spnbolizing twelve attributes, or Powers. Today the ancient system is in fragments; eachof the sciences,arts, and religions contairs remnants of it. Becausewe insist on a literal interpretation of past wisdom, it has been distorted and obscured. We make undue efforts to see "things" and people outside ourselves ashaving power over us. lrstead of discovering the principles of twelve-aroundone within our deeper se1f, relegate it only to a pattern of we holes in a salt shaker and a game of Chinese checkers. Yet we cannot'and should not expect to rediscover the full body of ancientwisdom by studying dusty monuments and myths fu1l of idioms and subtle referencesunderstood only by those who lived at the time. The perennial wisdom requires each individual and age to discover it anew in extemal mathematics, expressing it in ways and symbois suitable for those times and cultures.


Srvn l|rprnn

Virgi [nchanting But the seventh day is the Sabbath of the Lord thy God; in it thou shalt not do any work . . . -Genesis 20:10-1L Wisdom hath builded her house, she hath hewn out her seven Pillars -Pro.erbs g:7 If you keep a thing seven years, you are sure to find a use for it. -Sir WalterScott(L771-1832, poet,noaelist, historinn, Scottish and biographer) Measure seven times before you cut. -larpenters'


He found the vast Thoughi with seven heads that is bom of the Truth. -Maha Upanishad A 1iehas sevenendings. -Swahili prooerb

A regular heptagon cannot be constructed with the geometer's ttuee tools and so is not bom like other shapes through the uesica piscis.Btt an approximate heptagon is possible to construct.





Sevenis perhaps the most veneratednumber of the Dekad, of the number par excellence tlie ancientworld. Its name has been similaily pronounced around the world: in Sanskrit saptan,Egypttan sefek,Latin septem,Arabic sabun, Hebrew crT|Es cHotRsctRcLrNGs shna, Sparisln siete, and so on. The Greeks, whose word for CL,EANSINGSCLEAN.BEASTS COINS COMPANIONS or nicknamedit "sebo," "veneration." Seven sevenwashepta, CONSTELLATIONS CONTINENTS has always predorninated in mythology, mysticism, magic, covENANTS COWS CROWNS ceremony, and superstition (both lucky and unlucky). Its CURSS DAUGHTRS DEFENDERS DEGREES.OF"WISDOM DEYILS appeal endures to this day in religiorl nursery rhyrnes, and DOGS DOORS ELDERS EUNUCHS folk sayings, from the reverent "seven angels before the EYES FISHES FRAGRANCS throne" to the colloquial "seven-year itch." Our feel for GABLES GATES GENERATIONS GIANTS HEADS HEAYENS IJONST. sevennessis a measure of our sensitivity to the archetypal MEN HORNS ISLANDS I(NGS LA(S principles of the Heptad. LAMPS LEADERS LETTRS The Bibie, literary source of Western culture, contains LIATIONS LOAVES MATRIARCHS thousands of references sevens,heptarchies,and septeto METALS MOUNTAINS MYSTERIES NATIONS OATHS OYRSERS PATHS naries, the Greek and Roman terms for sevennesses. Seven PATRIARCfIS PTALS PILLARS were in the den with Daniel, and seven locks of Samlions PLAGUES POOLS PRIESTS son's hair were cut. Joshua'sseventrumpeters circled JeriPRIESTSSES PRINCES PRINCESSES PROSTRATIONS cho seven times. There were seven voyages of Sinbad and OUEENS RAYS RPIITIONS sevenWonders of the Ancient World. Sevenperambulations RIDGES RIVERS RUNGS are made around the Kaaba at Mecca.Animals considered SACRIFICES SAMURAI SAS SINGERS SINS SISTERS SLEEPERS cleanwere admitted to Noah's Ark by sevens, unclean aniSNAKES SNEZES SONS mals by twos. The seven"liberal arts" of medieval education STAIRCASS STARS STPS STONES were so named becausethey representedseven paths of STRAMS SWANS TMPLS THUNDFS TREES TROUALES leaming intende d to liberateus from a mundane liie. Snow TRUMPETS VEILS VgNGEANCES I /hite and the SevenDwarfs is an old tale containing hidden YIALS VIROINS VIRTUES VOICES wisdom within its symbolism. You probably knolw many VOWELS VOYAGES VARRIORS more references.\A/hat is the source of our fascination with VATCHERS WIS.MEN WISE.WOMEN V/OS WONDERS YARS seven?What are the principles of the Heptad for which we accordit such specialattention? Sevenis on iveryone's lips at one time or another. When asked to think of a "lucky" number, you're more than likely to choose seven or tfuee. Lottery games often disallow the number 777to prevent excessive payout due to its popularity. Think of something familiar that comes in sevens.Some Hail! Great Mother,not people recall that the opposite sides of honest dice add to hathbeen uncoaercd tW seven, but rnost think of the seven days of the week. Unlike the Monads of the day, month, and year, which birth. are measuresof the motions of the earth, moon and sun, the -Egyptian Hy'rnn to Neith ancient measure of a "week" (from the Gothic utiko, signtfy-



ins "sequenceto which we come," which derived from the ffrptiair rak, the word for "festival") is not basedon any cycle, although it is cgnveniently closeto being a "Jt"idut fourth pait of the moon's monthly cycle,of29'5206days.(ong the concept of weekdays and weekend "-oo.tth";. \44ri1e grew out of the Industrial Revolution, there is something ur is that moves with the seven-dayrhythm. Sevensomehow a creates familiar cycle. In many religions every seventh day is a Sabbath (from from labor"), a sacredday dis" the Hebrcw shabhat,to cease tinct from the six-part structure-function-order of the worldly weekdays. The weekday is both sacred and profane, light and dark, God-made and man-made, timelessness wi-thin time. Bui on the Sabbath there is only holy and holy; there is only light, as nothing else is attended 1., DY"lity is suffused into the realm of the eternai. The Sabbath is the virgin bride of the week. Traditionally every seventhyear was a "year of release," when a field was allowed to rest from planting, when debts were forgotten and slaveswere set free.Today,the law sets sevenyeirs as the duration of the statute of limitations for somecrimes.Folk wisdom refersto life proceedingin sevento year periods. Consider your own life. Doesit make sense lee yorr. development as stages between the ages of zero-seven-f ourteen-twenty-one-twenty-ei8ht-thirty-f iveforty-two-and so on? A group of seven comprises a comPleteunit, a whole event. But a group of sevenis different from other wholes we've encountered, particularly the Monad, Triad, and a Hexad. The Heptad expresses complete event having a beginning, middle, and end through seven stages, which keep repeating. Seven representsa complete yet ofiqoittg process,a periodic rhythm of internal relationships. All configured efforts are 1edin sevenstagesto perfection.To look deeperinto the principles of the Heptad, we have to look at the unique arithmetic and geometry of the number seven.

a AII theworld's stage And all themenand womenmerelyphyers; Theyhaoetheir exits and their entrances; And oneman in his time playsmanyparts, ages. His actsbeingseaen -William Shakespeare

MATfIEMATICS THE VIRGIN OF The number seven occupies a critical place within the Dekad, where it acts as both a link and a chasm. As a Iink 1x2x3x4x5x the between firstsix and lastthreeterms,



We encountered seven sections formed by Borromean Rings, and as six circles around one, but not as sevensided figures in nature. Only SevenNumbers exist. The ancient philosophers did not consider one, two, or ten to be actual numbers but rather their source and resulf so the Dekad contains only seven nurnbers: three, four, five, si; severy eight, and nine.

6 x 7 equals7 x8x9 x 10 (equals 5040). a chasm,with As seven absent,1 x 2 x 3 x 4 x 5 x 6 equals 8 x 9 x 10 (equals 720).I4trhether value of seven is present or absent,its the location servesas a pivot baiancing the ten. No other number or position within the Dekad doesthis. The ancientsreferredto sevenasthe "virgin" number. It is untouched by other numbers in the sensethat no number lessthan sevendivides or entersinto it as two divides four, six, eight, and ten, three divides six and nine, four divides eight, and five divides ten. Sevenwas also consideredchildless since it produces no other number @y multiplication) within the ten, as two producesfour, six, and so forth. To the ancient philosophers,myths were like our scientific and mathematical formulas. The gods and goddesses rcpresented principles wilh certain properties and athibutes associating them with mathematical archetypes. The relationships betweengods and goddesses correspondedto the relationships among numbers and shapes,arithmetic and 8eometuy. The mythology surrounding virgin goddesses(Neith, Athena, Minerva) often elucidates the principles of the virgin Heptad. The myth surrounding the "birth" of ihe Greekgoddess Athena, or Pallas Athena Parthenos,was explained with the construction a heptagon. of Athenapresidedover war and wisdom, settled disputes,upheld cosmic law; she was first to teach the scienceof numbers, the arts, and trades; sheinvented cooking, weaving, spinning, the plough, ship, flute, pottery, and chariot; and shewas patronessof the city of Athens. But she was not born in the ordinarv wav through a womb. $/hen Zeus was told that one of his chiidren would overthrow him, as he had usurped the throne of his father Kronus, he swallowed his pregnani wife, Metis, goddessof measure,mind, and wisdom, whole. He soon developed a fierce headache.\Atrhen blacksmith the deity Hephaestus split his crown with an ax, Athena, mature and ful1y clothed for battle, with a mighty shout sprang forth from the cleft in his head. Sheremained virgin through life and, of course,bore no children. The orieinJof her nameareunclear, "A-thene" is thought to sidlify -I but have come from myself" or a-thanos, signifying ,,deaihless,,,



"eternal." The birth of Athena is depicted on the eastPediment of her temple in Athens, the Parthenon, over the itself. entrance mathemafics,and art were Mvthology, religion, science, We of system philosophy' cangain integrated oncefart of a-n by looking at dre namesot the gocls insigirt into this system in and"goddesses their original language'In the ancienttraditio"ns of Greek symbolii mathematics the letters of the alohabet represent numbers. \A4renthe values for the letters of Ath"t u'" Gt""k narne are added togetherthey equal seventv-seven, a clear reference to number seven' The numerical value of the lettersof her epithet,Pallas("maiden"), combine to equal 343 (equals 7 x7 x7), the volume of a cube whose sidesare sevenunits long. The letters of Athena's title Parthenos("virgin") add up to 5L5.Five hundred fifteen is not a multiPle of seven' But (nearly 51.5) degrees measures the vertex 51,.4128571.... angle at the corner of a regular heptagon, the seven-sided by pol*ygo".Use a calculatorto divide ihe 360 degrees each Statue of Athena from the one of the ten, Acropolis displays on her bf thi values one through ten' \iy'hile every divides 360without remainder,only the seven- chesta gorgon head enclosed seuen, except sided polygon presentsan endlessdecirnal and an unmea- within a heptagon, associating sevennesswith the virgin. surable,elusive angle from its centerto its corners.Assiduously maintaining virginity, the Heptad cannot be captured, that is, made manifest, by arithmetic or geometry, the mathIt ematical creating Process. appearsto the geometeras an to un{old but not uniting with any suitor illusion, seeming among the many divisors of 360. It's well known that the regular heptagonis the-smallest polygon that cannot be constructed using only the three straightedgeand pencil, the tools of the georneter, comPass, mirror the methods of the cosmic creating the tools that process.In other words, an exactheptagon is not (and cannot be) "bom" like the other shapesthrough the "womb" of piscls. This explains why the virgin seven cannot lhe z.tesica be entered (divided), cannot produce numbers within the Dekad, and cannot be captured: simply becauseit is not bom. Use a calculator to divide each number one through ten of by seven.They each yield the same result: the sequence digits 1-L2-8-5-7 cycling endlessly, although they each



Among all the shapes,only the heptagon cannot be captured precisely. H


AYO 180 0 120 60 90 90 72 108

Center angle Vertex angle

360 infinite

Seven doesn't form finite relationships with other numbers. I4trhenarly number is divided by seven it always leaves an bndlessly looping trail of repeating decimals, L42857,my sterior;;slyomitting 3, 6, and 9. 7/7=0.142857... 2/7 =0.28571.4... 3/7 =0.428s71,... 4/7 =0.571.428... 5/7 =0.714285... 6/7 =0.8571.42... 7/7=1.000000... 8/7=1.142857... 9/7 =1,.2857L4... -1.428571 10/7 = ...

begin with a different digit. Six digits, like the six days of the week, are set in endless motion around the unseen Sabbath The common sayr,:;g"at sixes and sevens with each other" refers to seven's aloofness, its virginal unwillingness to mix with other numbers. A carefully conskucted mathematical canon underlies mythological reiationships. The archetypal principles of arithmetic and geomefry corresponded to deities in aiystem more widespreadthan is presentlysuspected, thafwove one together the arts, sciences,and religion of ancient civilizations. ln the ancient "mythmatical" canon,eachgod and goddesswith fus or her titles and attributes repreints a leiqth (77),an area(spreadby an angle of -51 .5 degrees), and a ;ir volume (343)-The appropriate units wete applied to designing two- and three-dimensional sacred objects, paintinls, rcliefs, statues, monuments, and temples dedicated to tliat deity. Temples to Athena, Minerva, and Neith were de_ signed and constructed around the number seven. This tradition continued in Renaissance music, which called for seven voices when singing of the Virgin Mary.

CONSTRUCTAPPROXIMATEHEPTAGONS AND HPTAGRAMS (1)Construct oesr piscis (2)asquare theupperhalf. a ca and in (3) Draw diagonalsto find the center of the squari-*d.lre




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The Bhth of the Hepiagon.

it to inscribe (4) a circle within the square. Theseare symbols of heaven and earth, Monad and Tehad. ('5) Note the two ooints where the small circle crossesthe circles beyond the piscis. aesica Open your compassto span thesetwo Points. This is the length of one side of the hePtagram. (6) Walk the compass around the circle, leaving a mark at seven points. Connect them to configure an approximate heptagon.



piscis bur pops out frorn-the top 6f tne ;;Ifiilil -ztesica beyond i! from the cror,r'nof the circle, the Monad, from the head of Zeus, het "father." You,ltfind that the fina I " step,,of the compassdoesn't quite match the starting point. A ,,t-rue,, regular heptagon cannot be precisely cons6cted with the geometer's tools; it always eludes us. Sevenis therefore a s)..rnbolof eternal rather than marrifest ,,things.,, Walk the seven steps in the opposite direction and ch-oose yout as actual point a mark in between the first and second set. (8) Connect every other point to make one type of hepta_ gram star. (9) Connect every third point to make anoiher f'?_e.oi (10) Both heptagram stars drawn _heptagramstar. within the same heptagon createsthe We6of Athena, which the goddess of weaving spins in her mythological exploits. at the hegtagram stars a while. Do y-ouget i sense . lye of their moJion? Glide your eyes along the iines'until you can move faster. A heptagram imparts a feeling of move_ menr Decause eye attempts to resolvethe unevenness our of ure torm. We senseits elusive etemal characteras motion. The number seven represents the most direct link with archetypal pattems accessibleto us in syrnboiic form. Constructthe Webof Athena.DoodlL with it. Connectits crossing points and shade some areas. Get a feel for the angles and pattems of severuress. Cut out the heptagory and creaseits lines on both sides. Fold it to create inieresting three-dimensional structures. I like to fold construction3 done on reproductions of ancient art to seehow the differ_ ent parts are composed and how they relate to each other.

LikeAthene, does emerge it not dtectly fiOmWithinthg


Heptagram stars

Web of Athena



Demeter' Unlike Mother goddesses,such as Isis and the hepby who are rePresented the "solid" earthy square' they are tueot ut uitgirr goddesslsdo not nourish us because .ro"tor"r"ttt"it q"tite the sameway' We glimpse only a sPecte.r of ihe virgin Heptad through our approximatio"t YlF number aid shape; she cannot be reached by worlclly means. Another method: Construct a triangle and square within piscisIo weave the IlrleTesica principles of Monad, Triad, and Tetrad.

MAKE THE HEPTAGON'S IPs RELATIONSfI AU DIBLE The Web of Athena is composed of only three lengths: the side of the heptagon und wo types of iength- between ooints. We carr-muke their abstract mathematical relationihip" ^ot" comprehensible by making them audible' Constrict a large heptagon on a piece of wood, and hammer threenails halfway into the wood at three Points a9 shoyun Tie and tighten a piece of guitar string or nylon fishing line arotrnd tiem. Piuck them and listen' Find their tone sequencefrom high to 1ow.Listen to the intervals they make, the jurnps betwedn them. Are the jumps the same -or differenti pluck each two shings simultaneously, and listen for the rhythrnic beat arising in their differences.

A7-ridrdyigu*i, to impossible draut c With perfe t mathematical prccision; And if you try to do it You sure areabsolutelY Tofind your ffirts treated with derision. are And yet there s philosopher who readilY declare That nothingin this world is reallytrue, And soI'ae drawna by heptagon triangleand squafe. it For any humanpurpose rpill do. -John Michell



APPROXIMATE +IEPTAGONWITH THIRTEENA KNOTTEDSTRING When the ancients wanted to construct a heptagon as the foundation of a temple they began with a loop of rope having thirteen equally spacedknots. To replicate this construction on paper tie thirteen equally spaced knots in a loop of cord or string. Form a triangle with sides of four, four, and five by stretching the loop arormd three thumbtacks. Use your straightedge to draw a straight line from the fourth knot on the longest side to its opposite comer. This makes one side of an approximate heptagon. Remove the two upper tacks, and rotate the friangle around the remaining central tack so that it continues where the previous hiangle left off. Again mark the fourth point on the long side. Continue to rotate the triangular loop and mark each fourth point on the long side until the sevensidesof the heptagon are created.

TIE A HEPTAGONAL KNOT A seven-sided knot can be tied in one long, thin strip of paper. Follow the path shown in the illustration. Carefully pnll the strip tight, pressing it flat, keeping the sides as equil as possible. Its underlying design reminds us of Borromian Rings, a trefoil knot, and an open lotus. Hold the completed knot up to the light to seethe Web of Athena within.

CONSTRUCT HEPTAGONWITI{ SEVEN A TOOTHPICKS By trial and erro4, out seventoothpicksasshown. Besure lay that the two toothpicks making the top comer angle aim precisely at the ends of the bottom horizontal toothpick. Four_ teen (= 2 x 7) of thesetrianglesrotated around a point rnake a beautiful, more intricate variation of Athena,s web within it. - Lgam to draw heptagons and heptagrams freehand. Doodle with them to beco-me attuned io tie principles the Heptad holds. Connect points and extend lines to iisclose fu rther heptagonalpatterns.



Arrange seven toothpicks. unsharpened pencils, straws, or nails to form this section of a heptagon.



Theseare just some of the many methods people have employed in pursuit of the elusivevirgin number.Just asno method using the geometer'sthreetools,which replicatethe "tools" of the cosmiccreatingprocess, light, energyand matter, will make an exact heptagon, neither can nature conskuct preciseheptagons. ,/ But precise heptagonb are not necessaryin natural or symbolic human constructions. fact, they can only be seen In "imperfectly." Heptagonsare most powerful when they are represented this way.

WeaveAthena's Web upon ClassicArt This painting within a Greek bowl depicts the virgin Kore, or Persephone,with Hades kidaapped into the urderwild. Seven dark crossesappear around the rim, giving us a clue to its design geometry. A Heptagon has been constructed around the circle with its comers halfway in between the sevencrosses. place tracing paper over the picture, and connect the heptagon,s comers to produce the regulating lines that composethe scene.



LYRE ENCHANTINO The math and myth of severythe Heptad, are intimately Bothhavein comto related thoseof twelve,the Dodekad. iriangleand square. mon the interplayof Triadand Tetrad, That is, 3 + 4 = 7 while 3 x 4 = 12. These simple equations may not seem like much, but they are responsible for myriad cosmic structures and relationships. We saw how the twelve-part geometryof the squaredcircle (chaptersix) with twelve moons around a thirteenth at the center holds the canon for the structure-function-orderof a settled society. Within it are heptagram stars representing a different kind of structure-function-order, that of the wandering, nomadic iife. While twelve has relations with all numbers within the Decad, sevenhas none. Sevenis the hidden, unborn, eternally elusive side of twelve. Sevenand twelve are associatedso often in myth and legend that it becomesobvious that they refer to a systemof symbolic philosophy. It was noted earlier that twelve vultures showed Romulus sevenhil1s on which to found Rome. The Greek pantheon of twelve deities was representedon earih by the seven peripateiic wise men. Most widely noticed was the associationbetween the seven nomadic planets(from the Greek for "wanderers") whose courses change periodically against the backdrop of twelve fixed zodiacal constellations. the Kabalah,the twenty-two letIn ters of the Hebrew alphabet are divided into three groups, of three, seven, and twelve, representing, on one level, our deeperself, the solar system,and the galaxy. Seven-twelvealso describesthe essentialrelationshir: between the modern seven-notediatonic and twelve-note chromatic musical scales. The chromatic twelve-scale includesthe diatonic seven-scale plus five sharps or flats between them, giving the scale shadings of musical color ("chroma"). C C # D D # E F F # G G # A A # B The modem piano of eighty-eight keys displays seven and one-third chromatic scaleswithin ihe approximately ten octavesof human hearing. The Chinese, Polynesian, and Scottish five-note penta-

Geometry of the twelve-part celestialcanorL as the piano keyboard, relates to sevenness since3x4=12and3+4=7.



\t\tLether the cosmos is represented as a musical scale as in ancient times or as the modem sonic ald electromagnetic spectrum, they both depict a universe based on vibration.

yearning, is made of five of thesetwelve notes. The people ofall higher cultureshave felt deeply that the world is ordered. In ancient times the design of a musical scalewas part of a sacredphilosophy reflecting the world's inherent order,its tonesnot chosenaccidentallyor purely by ear.The musical scalewas anotherwav to representa model of the r,rniverse and served as a tool for en}incing peoples' harmonious relation with it. Westem civilization's seven-note diatonic (from the Greek "acrossthe tones") musical scale (the piano's white keys) has been used from time immemorial. In ancient times it was traditional to arrange the strings to play the scale downward, asif it were descendingfrom heaven.The modern names of the sevenfamiliar notes in descendingorder, DO-SI-LA-SOL-FA-MI-RX-DO, were proposed by Guido d'Arezzo, inventor of the musical staff, around 1000 A.D. Thesepopular namesareonly the first lettersof Latin words whose translation revealsa cosmologicalstructure derived from an earlier age: DOrninus "Lord" Absolute Slder "Stars" All Galaxies LActea "Milk" Milky Way Galaxy SOL "Sun" Sun FAta "Fate" r-lanets Mlcrocosmos "Smalluniverse" Earth REginaCoeli "Queen of the Heavens" Moon DOminus "Lord" Absolute

tonicmusical scale, whichmosteasilyproduces feelof ihe

J eedeep enougbandyou see musically; heartof the Naturebeingezterywherc The seven-notescaleis meant to model the hidden side of the macrocosmicdesign, the universe ruled by rnathemusic,if you canonly matical harmonies of music. The scalestrucfure impiies that rcachit. -Thomas Carlyle

the universe emerges from absolute divinity, descends through a seven-stage celestial hierarchy, and refurns to absolute divinity. In order to create music the string is bowed back and forth by the motions of the generative Dyad between the extremes of harmonv and discord, love and skife, light and shadow. The ancientsdesigned and used musical scalesto play the harmonies of the heavens, the music of the spherei



Frequency in cycles per SeCOnd 1O3

4X 1014 I x 1014 1Or2 \ /1Or5

\noo.= Yellow i Blug Green

sPectrum. Electromagnetic pleasing to both gods and humans. Music was meant to iUo* ti" higher frinciples to enter our lives tfuough our senseof hearing and our emotron. Musical strings were considered imperfect representations of archetypal ideals and pure mathematical rnotions' \ trhenvibrating, these strings were thought to resonatewith the archetypal tones, manifesting cosmic principles. Thus, music wai ieen as having great power for producing harmony on earth. Music of the lyre was used to accompany and enhance the voice. But it was part of a larger administrative and therapeutic system involving colors, fragrances, flavors, textures, herbs and plants, minerals, metals, myths, agriculture, meteorology, and astronomy. It was well known throughout ancient civilizations that music bypasses the intellect with the Power to manipulate passignsand emotions.In this way music was used to heal. As we find by being very quiet within and observing ourselves, sound has an effect on our inner substance. If we were more aware of the effects sounds have uPon us we would be more careful about the frequencies we exposeourselvesto. Healing or therapeutic music was attributed to Egyptian, Babylonian, Indian, Chinese,African, Native American, and Persian priests, to Pythagoras, and the myttrical Apollo and Orpheus.



Apollo ("moving together"), brother of Athena and the only major Greek deity with no Roman counterpart, was the Olympian god of harmony. Everything he was involved with displays order, rneasure, and beauty: music, poetry, prophecy, medicine, art, archery. According to myft, he acquired the lyre from Hermes and with it taught the principles of harmony on earth. The Three Fateswere said to have invented the first five vowels, but Apollo was said to have invented the other two vowels (long o and short e) so that his sacred lyre would have a vowel corresponding with each of its seven strings. His son Linus took it further and invented rhythm and melody. Apollo presented his lyre to Orpheus, who, myth tells us, used it to enchant people, plants, animals, and the weather. By its tones Orpheus guided his wife, Eurydice, Apo1lo from the darkness of the underrvorld into light. He is said to have made trees and rocks dance to his mirsic throughour Greece,where they remain to this day in the merry pu[u.rrs he left them. lythagoras is said to have calmed a raging bear and soothed the murderous passions of a drunken man by playT ln Egyptthepriests sing ing appropriate tunes on the lyre. In ancient times the design of a musical scale involved hymns to the Godsby the science of "mediatioru" the binding of two ,,opposite,, utteringtheseoen uowels . . tones by a third tone placed harmoniolsly between them. in succession, soundof the Harmony (from the Greek hnrmonia, signifying ,,fitting which produces strong a together") in music, as elsewhere in the as cosmoi, colmesfrom musical the reconciliation of opposites by a third element, bringing irnpression on them all a new unity. The ancients saw that music thEorv their hearers if theflute as holds the principles of social harm ony and discord. and lyre wereused. . . but Numbers themselves arise from opposites, from the perhaps hadbetternot I interplay of Monad and Dyad. Geometric shapes emerge enlargeon this theme. from opposing circles of lhe aesica piscis. tone is creatEd A when a motionless string is plucked by a moving finger. In -Demetrius (First century Greek myth the goddessHarmonia was bom from oppo_ 8.c., Alexandrian sites, her mother Aphrodite, goddess of love, her fatirer philosopher) Ares, god of war._Harmony is their balance. Through music the ancient philosophers could hear the relaionships among the gods and goddesses.



Sound is simply vibratiory but the agreeable and disagreeablesensatibnsproduced by music are due to. tones bEingcombined harmoniously or discordantly'Cos uc narmonles plaved with earthly shings bring the archetyPal principles to earth, thus making music therapeutic' ' The mosi famous musicosmic scale, the one Western civilization usestoday, is derived from the scaledeveloped and taught by Pythagorasbefore 500 n.c. Undoubtedly he leamed"the myiteries of music in Egypt, Chaldea, and lndia, but he wis the first to teach its principles outside the temples. Pythagoreanscalled the harmonious cosmic relationihips the "music of the spheres" and taught that music is based on the mathematical relationships, found in nature, and can be expressedby ratios of simple whole numbers. If you have a guitar or other sfringed instrument, or even a rubber band stretched between points, you can try these experiments to replicate the cosmic harmonies and forces in the safetv of vour own home. First, pluik a string of any length, thickness, and tension. The string is a Monad. Al1 subsequent harmonies begin and end in this unity. Listen to the tone without narning it' Observe the shape of the vibrating string. It resembles a piscis, a womb whence sound is bom. But is sound ztesicn really born? Sound is a transitory wave of air and doesn't stay long, just asthe elusive heptagbn isr/t rcally constructed. Every structure in the universe is attuned to a level of the cosmic scale.Tap any object and listen to the tone it makes' It's telling you its natural name. Each configuration has its own note, its vibration by which it sings to the cosmos. Now estimate by eye (don't count frets) where you think the exact middle of the string is. Press (or put the bridge) there. Pluck each "halJ" length and listen. You'll know if you've {or;nd the exact rniddle becauseif you have, eachhalf will sound the same.Now find the centet by listening, moving your finger until both halves sound identical. Most people can find the middle of a string much more accurately by ear than by eye. Pluck the entire string agairy and you'll recognize it as the same note as each of the halves, but lower in pitch.

On thisaccount kePt Pythagoras a IYre music with himto rnake and to sleeP going before in uponwakinS, order his to always imbue soul qnalitY. with itsdioine -{ensorinus (Third cmturY e.o., Roman scholar)

Pythagoras performing the first experimmt in physics, the mathematical tuning of the musical scale.The weights of the strings result in different tensions arrd tones. The numbers representing weights can just as well be the lengths of string with idenfical tension.


Vibrating string lengths represent the fundamental tone (C), its octave(c) , and the harmonious fifth (G), the "chord of triumph," between them.


is a secret

arithmetic exercis and al e theperson who indulges in it does realize not that heis manipulating numbers. -G. W von Leibnitz

_ These opposites of fuIl string and half string define the bounds of the cosmos.Monad and Dyad are the parents of all tones just as one and two are the pirents of number, and the_circle and aesicapiscis give 6irth to all geometry. Although the sounds produced by whole and hilf-string sound different, they are alike, being one octave apart. But unless they form a relationship in betweenthem they produce nothing new, always the same tone, the One. Harmony is the child born betweentheseopposites.She appears on the string at the point that divides it into two unequal parts so that pluckingeach part produces same the note as theotherpart,but different in pitch, an octaveawav. Find this dividing point by moving your finger (or bridgej along the sking, plucking eachpart, andlisteningfor the"same noie arisingin each, high and one low' This is the tone of con_ one cord between the Monad and Dyad that the ancients sought as the foundation on which to build the musical scale.lnt"er_ nal self-agreementis the essenceof harmony in music, nature, and human affairs. . ly]hagoras found this "balancing', point at two-thirds the full length of the string. A divi;io; here creates two parts, one of which is twice (two-thirds) the length of the other (one-third). The two parts sound the sam"e eact as other,also an octaveapart. They replicatebefweenthem_ selvesthe doubling relationshipof their parents,the Monad and Dyad. This division into ,,thirds,,introduces the Triad, a birth, a child of the essential unity different from its nar. ents yet in harmony with them. Ptl* the full string.then two-thirds of the string,and ,. listen. No matter what the original tone, its fwo_third.-s tone has a certainfeel to it, in relation to the whole string. It is tra_ ditionally known as the ,,chord of triumph,, for tf,e insoir_ -We ing way it moves_everyone who hearsit. respondnoi to the tones themselvesbut to their relationship, their musical interaal. By continually subdividing each two_thirds section by another two-thirds we create a sequence of shorter string's and higher tones, each building upon and maintaining tfie same harmonious relationship with tl-re notes before"and after it, always the chord of triumph. If you are careful vou 'the will find, as Pythagoras did, that after seuenth diviJlon


239 Pythagoreantunhg. Endless scalesian develoPas "fifths" (clockwise) by continua\ dividing a string at twothirds (or extending bY {ourthirds); or as "fourths" (courterclockwise) bY continuallY dividing at three-quarters (or extendingby three-halves).


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the notesrepeatthe earlier cycle,but sharper.He also {ound that after the trnelfth division the original note was virtually sounded again, although slightly flatter. These twelve tones spread acrossseven octavescomplete the cosmic picture, Theseare of although the sequence notesis not alphabetical. the twelve notes of the piano's chromatic scalerepresented by its sevenwhite and five black keys. They lead us to the enchanting virgin. Her song is the mathematics and physics of music itself. Tlds process of continually dividing a string by twothirds to produce new higher tones is called the "circle of fifths," although it is more like a "spiral of fifths." "Fifths" refers to the fact that each subsequent division at two-thirds produces tlrLefifth note ahead in the rnusical sequence A-B-C-D-E-F-G. In other words, if a length of string soundsnote A when plucked, then two-thirds will produce a higher note E; two-thirds of that will sound a higher B, and onward, always five letters ahead.The fraction two-thirds was so important to the ancientEgyptians,who successfully integrated music within their civilization, that they honored it as the only fraction to have iis own gl1ph. A11other frac-

The musical "fourth." Dividing a string at three-quarters of its total length produces the "fourth" note ahead in the scale.Continually subdividing into three-quarters produces the same twelvetone scale as the spiral of "fifths" but lessrapidly and in the opposite direction.



tions were denoted by sums of srnaller fractions having a 1 as the numerator depicted by the "mouth" glyph, like a vibrating string, over the numeral (e.g.,3/4 -- 1/2 + j,/ 4). When the same 'ispiral of fifths" is looked at from the opposite direction, each higher tone is the fourth le!1er ahead. The same scale but ascending, called a "spiral of fourths," can be produced by continuou sly extendingthe string by four-thirds, or by dioiding it at its three-fourths length. A tone's "fourth" is the inverse ofits "fifth." On the spiral, the fifth note ahead is the same,but an octavedifferent, of the fourth note behind. A strinq's fourth and its fifth added together comprise an octave,"awhole comprehending both harmonies.The tone, octave,fourth, and fifth hold the mathematicsthat builds the harmonious musica scaIe. This was the method of turring instruments used from the time of Pythagoras through th; Middle Ages and into the Renaissance. produces more concord.s It among its constituents than any other selection of tones. Musical instruments tuned to the ancient Pythagorean scale sound brighter, as an old frescolooks when cleanedof centuriesof grime. Some native peoples around the world and even sorne modern rock groups tune their instruments the anclent way. Thesetwelve notes were the onesheld by eachchoir of the twelve provinces of classicsettled societies.The spiral of fifths representsthe zodiacal year in twelve phases,twelve processes life, twelve spiritual attributes. of Musicians eventually noticed that the thirteenth step shouldbrtng us back precisely to the starting note seve-n octaves later, but in reality it is slightly flai. The actual vibration-isonly 98.65603. . percentof the archetypalideal, . a tiny difference termed the "comma of pythagoias.,, The ancients. knew _this tuning to be very powerful, literally enchanting and therapeutic. But wher orchestras deveioped in the seventeenthcenfury,musicianswanted to tune their instruments precisely with each other. The problem with the ancient tuning is its difficulty modulating Lefrteen keys. So the comma of Pythagoraswas distributid evenly among the twelve notes, flattening each by one-forti_ eighth. This system, which tunes the modern piano, is



aesicapiscis.Extend the diameter to meet the cirof the new circle at the length of string "rr-Ier"nce whlch sounds the lower note E, a musical " fourth" from B. 3. Draw a vertical line between the Points of the piscis. Center your compasswhere the line aesica the crosses diametei and open it to point E Tum a both the smaller circles' circle to enclose 4. With the compass open to the same radius and centeredat poini E, tum a circle and createa larger uesicapiscis.Extend the diameter to determine the length of string whose tone sor;nds the lower note A' 5. Repeat steps (3) and ( ) four more times creating pisces which give the larsei and larger uesicas for ten"Aths lo'i'er notes D, G, C, and F If you wish the c*onstn:ct longer lengths for the full twelveto note chromatic scak!,continue the Processto create A;, notes 86 E6, Q, and G6.



o Each larger circle extends the previous diameter by onethird; that is, when the scale is descending, each string is four-thirds its original length. Musicians call this interval a F or tltetrueknowledge "fourth" since each next note is four letters ahead alphabetof musicis nothingothn ically. When built in the other direction as an ascending scale,each note is a '.fifth" ahead. Ln either casethe result is tlan this: to know the the samespiral. orderingof all separate Gaze at the completed geometry until it seems to thingsand how theDiuine becomea bunnelof expanding uesicas pisces. Reason distributed has \Atrhen these lengths of string are plucked sequentially or them; this orderingof simultaneously, eachof the twelve tones is harmonious with for those directly before and after it. aII separate thingsinto If the geometry on this page were audible and we could one,achined by skilful hear thesetones asthey are constructed, we would hear them ruson, makes sweetest the #owing rapidly lower. These twelve tones span seven and truestharmonywith octaves.We need to gather them into a single octave within the reach of the human voice. This, too, is done by geometry, theDiztine Song. lowering notes by doubling short lengths. Every doubling of -Hermes Trismegistus a string's length lowers the octave but maintains the note. We will use this technique to bring all the notes into one octave,

Thesesevennotesspanthree octaves.


245 The word "music" comes from the Greek mousike ("muselike"), which derives from the Eglrytian gfiPhmes ("birth, generation").

in an alphabetic sequence within the last circie we drew' 'ancient Greeks recognized seven modes of musical The scale, each a sequence of twelve notes along the spiral of fifths (or fourths) but each beginning with a different tone' Ttre modes were said to resembleseven human "moods" and would be played for differ'ent therapeutic and social purposes.The Dorian mode was consideredmasculine,the ionian feminine. Each mode also had a different architectural representation known by the same name. The Pythagorean sequence, the Dorian scale, b"gins with note E and descends:E-D-C-B-A-G-F (E'). To refashion our construction into this sequencewe must double different strings' lengths once, some twice and some even three tirnes. To double any length, just open the compassfrom the end point where all circles meet to its mark and walk this length along the 1ine.The placement of notes F and C need not change. Double the lengths for notes G and D. Do two doublings, or walk four lengths, to lower the notes A and E. And double, or walk, the shortest distance lePresenting note B eight tirnes. This will finally produce the ancient sequence of string lengths for a Pythagorean scale,a vibrating replica of the relationships within the expanding sevenfold cosmic strucfure.

Thrrc i, gro*rtryin the humming the of strings. -attributed to Pythagoras

Doubling the lengths of shorter strings retains the note but lowers its octave, bringing all notes within the same circle and octave.




Various woodwind arrd stdng instruments like the flute, panpipes, and Celtic harp can be designed using the proportions emerging from this construcfion. E D CB

DISCOVERMUSICAL HARMONIES PAINTING IN Botticelli's painting of Venus depicts the Roman goddess of love, beauty, and affairs of the heart, the Gree"k soddess -Venus/ Aphrodite, mother of the goddess Harmonia. Aphrodite was not bom in the usual way but is shown here in her mlthical habitat, arising from tl.refoam of the sea. Botticelli, like many other artists, sculptors, and archi_ tects over thousands of years, composed his works using harmonious proportions of the musical scale.ln this wai their paintings, relefs, statues, and temples were musi'c made visible, as pleasing to the eye as mr]sic can be to the ear. hr the case of temples and cathedrals, specific sacred




G# A#

h1"mnswele sung in keys associatedwith that deity or saint resonant with the architecture. This painting, a golden rectangle (seechapter five), is divided ilong its perimeter into the tones of a full musical octave (E to E) and a "fourth" beyond it (B). Botticelli particularly ernphasizesthe concords of the "fourth" (G and E) and "fith" (A). To discover for yourself how they guide the placement of figures in the scene, place a piece of tracing paper over the page and connect the rim's emphasized points with straight lines. Note how a line from the C above to eachbottom corner divides the canvasinto three triangles containing the winds, Venus, and n)rynph. Connect the significant points in other ways to explore the painting on your own.



SYSTEMS CRYSTAL SEVEN If you wish to employ the Heptad's principles when constructing a universe, you should not exPectto create any seven-siied obiects.Fbr example, among the myriad crystals,from quartz to calcite,lead to gold, none aresevefi-stded' canThe reasois are simple. The heptagon'sunborn angles. not completely coveJa flat surfaceor fill three-dimensional ,pu"u *itho.ri leaving gaps, so nature has no use for it as structure. Instead,the Heptad pokes into the world asa seven-step ol process,sevenindependentstagesor asPects 1.*191". ' crystals,but the Heptad There may be no seven-sided as appears sevenmaJorgrouPsor "systems"into which-all are Their dif{erences due beautifuljewelsbelong. o'fnature's to seven different possible relationships among their axes and angles. Cubic crystalshave three mutually perpendicular axesof equal length. Hexagonal crystals have four axes, three of which are of equal length and lie on a Plane with an angle of 120 degreesbetlveen them. The fourth axis is perPendicular to the three and can be of anY length.

Tehagonalcrystalshave three mutually perpendicular axes,only two of which are equal length.

Orthorhombic crystals have three mutually perpendicular axes, each of different length.

Rhombohedral crystals have three axes oI equal length but an angle of other than ninetY degreesbetween them.

Monoclinic crystalshave three axesof different length, only two of which are perPendicular

Triclinic crystalshave three axesof different length, none of to which is perpendicular any of the others.



beginning with the cube' univerLook at thesepich.rres, I he of sal sr.,rnbot "eaith" and the "solid" Phaseof-matter' crystals cube is the basic volume from which the other W" usually tend to notice a crystal'ssmool l?t"": lts tn-ree "-i*a. But look at the cube from the center outward' See eqlal length'J-oof lt th:l p"ip""dicu1ar axesof *"*Jiy you ian?eelhow spaceextendsalong eachaxis to manuntil around the sevenano ifest the volume. Co counterclockwise changview each this sameway, getting a feel for how the their lengths and angles' ing relationshiPsamong the axes' t tit" i"ven dif"feren possiblesystems ln eachsysa.i".-i"" {orming a progressionoJPlo' relationshipchanges, tem one portions among the lengths and anglesof their axes'In the cube three u*ei ute the same length and PerPendlcular'm are the tetragonal system only two are the same' Next/ none Then the tft" tu-J t""gth, but they are still perpendicular' aneleschangi, until sevenpossible "systems" occur' seven possibilitiesof atomic lacework' clo-se-packin-g in Seien stiges, or distinct steps, can be seen els-ewhere, nature,especiallywhere a doubling takesplace' We tiy iht: in the musical scile's seventonesoccurring betweenthe full length of string and its half. An example as close as your bodlyis the prolessby which it grows, the division of one cell inio two. This processof mitosis (seechaPtertwo) ls srmPly a transformation along sevenstagesor intervals' \ /hen the eighth step is taken, the octave is achieved and the one cell doubles to becometwo' Gambling, like warfare, has always been animpetus tor the development of mathematical ideas' Mathematicians and compuier scientists studying the possibilities ryiqT.u deck of cirds have found that when the pack is shuffIed it is simply being split into two parts, which are,recombined tnrouln inteireiving. They asked, "\n/hen is a deck of cards compl-etelvshuffled and random so that all cards have equal probability of being dealt?" Mathematicians have found ihat . . . you guessed it, seoenshulfles is iust right to create randomness.Fewer shuffles don't mix the cards enough, and more than seven shuffles is superfluous. Seven steps again completea whole' Seven-itep rituals appear in every culture and religion' Reverence foi this number, from the Seven Stations of the

in N aturedelights the nurnberseTJen. -Philo Judaeus 20B.c.(c. Alexandrian c. 40 .e..o., nhilnennher)



Crossto the sevenbasicballet steps,has occurredindependently around the world and across time. Natural exiressionsof the Heptad arenot as obvious as the results of other archetypalnumbers and shapesbecausethey must be seen as a sequence relationships.But they can be discernedif of we can leam to see,hear, and sensedifferently, to recognize the processes are part of, especiallythose sent to us in we elusive packages.

LIGfIT fANTASTIC The most elusive expressionof Heptad is a rainbow. No more enduring than vapor and light, the Greekspersonified this passinghue as the messenger goddessIris on a mission for Zeus leaving her varicolored trail across the sky. \n/hen sheis not in our sight we carry with us the daughtu. of Thuumatos (Wonder)as our iris, the colored part o1 our eye, the organ that perceives the wonder o{ Iight and color. Around the world, in religion, myth, and legend, light is considered the purest expression of divine priniiples aicessibleby our senses. Sanskritword for ,,angel,, deaa,litThe is erally "shining being," which gaverise to both ,,divine,,and "devil." Light has always been a symbol of greater understanding, a link between the eternal and the worldlv. That,s why the rainbow has been depicted i. -yth as a road or pathway, a bridge, an archerG bow, a divine necklace,a good snake,and a promise of divine beneficence. The wonder of sunlight and starlight is that it sets the universal speedlfu:dtat 186,282 milesper second,fasterthan anything else, traversing trillions of miles in absolute silence,yet it can be stopped by a mere outshetched hand. There'sno impactsincethe mais of a photon,a ,,particle,, of light, is zero. In fact, a photon ceases exist wlien it stops, to sin_ce energy is absorbedby whatever it hjts. In a sense, its light isn't even there. Yet upon this ioom of starlight all matter precipitates. _ The cosmic creating process,representedby tire equation E=mc2 and s)..rnbolizedby the geometeris three tools, describeshow light "thickens,, into archerypalenergy pat: terns that further condense into configuraiei matter"irithe fonn of the energypattern (seechaptei four). But a rainbow

W Iris



It never reachesthat most dense stage of materialization reveung remains perceptible yet ever intangible, brietly A rainbow snows itself onlv asa ievenfold pattem of energy beiore us the underlying pabtem,the frame of the structure '--fnu solid walls are Put on. pattem is not limited to sound' crystals' """"*ifdis characteristicof all vibratory phenomena' and lieht but perception(sight,hearing,taste,smell, toYth) tt-I AII serise spires by'means of our registry of vibration and-as sucn bu the harmonies inherent in the seventold mathe"tid"r we hear as music. matics The ret ning of the ancient musical scale into equallyin tempered.notes"came about during the same period which IsaacNewton performed his famousexperim*! yith a glassprism in a darkenedroom with only a narrow slit for sitrfinru to enter. To Newton's delight a shaft of sunlight enterid the prism and energed as a rainbow or chromatic spectrum spreadacrossa wide swath' ' Sunllght is "whote" in thai it is a mix of many vibrations overlapping. Some waves are very long. and some very short. Emer"gingfrom the sun they all blend into white' Sr-rniepresentativeof the Monad. The glassprism lieht is a uni"W, re"fractsthe t'uttlight, bending and spreading out its rnixed


waves like a deck of cards spread acrossa table. Like Athena bom from Zeus's head, the seven colors of the rainbowred, orange, yellow, greery blue, indigo, violet-appear to spring forth "fully clothed." Red's waves are longest violet's shortest. Just as the intervals between the notes in the musical scaleare not always the same,the color swaths of a rainbow are not of equal width but are proportioned like musical notes. Notice the different widths of color the next time you seea rainbow in the sky or around a garden sprinkler. Many people have seen rainbows and some know the acron)rm for the color sequence"ROY G. BIV"; but few peopie know whether red or violet is on the top of the rainbow (don't take my word for it but check for yourself that it's red). A rainbow will only appear to yo:u if you'rc tacing away from the source of light. The sun shines from behind and over you, does a doublereftactron within atmospheric water droplets, and the solar vibrations splay and wave into your eyes,where they are intetpreted as color. We claim to see rainbows and vet, like "constructed" heptagons, they're not really there; they don't manifest themselves"at" any location.Peoplestanding apart wili see the samerainbow in different places.Even your two eyessee two separaterainbows. Verify this phenomenon by viewing a rainbow with only one eye at a time-it will seem to shift. Walk, and the rainbow will seem to fo1low your steps. Try as you will to find its end, it will elude you, etemal virgin that it is. The rainbow appearsas an arc,but it's part of a full circle, which would continue underground if it could. The Greeks saw this as lris' abilitv to travel between heavenlv Olympus and the r:nderworld. The rainbow provides a transitory glimpse into etemal principles. Light touches everything. Anything can stop it, yet nothing can pollute it. Sincethe rainbow seemsto have infinite shades,why do we seeexactlyseaercolors?As usual, ihe answer lies within ourselves. Around 1800, the English musician-turned-astronomer William Herschel put a thermometer in the path of each color of a spectrum in a dark room. He gues;ed correctly



DEscax.?Es' Drac*Ara oF tdE RNNBow (Fod il. ld !"'tn ol l6lt')

that they would have different temperatures' But to his surjustbeyondeachendof the prise,the "empty" colorlessspace temperaturethan any of the colors' ioectrum had a higher By thus discoverin! infrared and ultraviotet lighthe showed that the spectrirm ii a continuity of which we register only a ,,arto- s[c" we label "visible light." But visible light is part of an endless range of vibrations extending to infinity at eachend. Herscheldiscoveredwhat moths, birds, bees,and other forms of life already knew: there are vibrations beyond these seven, Astronomers tell us that most of the universe (over 90 percent) is invisible to our eyesbut can be detected with equipment sensitive to radio waves, microwaves, X rays, ga-mmarays, and different wavelengths for which we have other labels. The clue to the rainbow puzzle is in the relative size of their waves- Eachwave of light our brain interprets as " red" is 1/33,000of an inch long. Violet's waves areabout half that atl / 67,000 rncl..Much asplucking a string and then another half its length produces musical notes identical in tone but different octaves, we seered and violet as the first and seventh "notes" of a "scale" iust before reachingred's full doubling or "octave" wavelength past violet' Just aswe can hear about ten octaves due to the spiral cochlea of our ear (see

Sunlight refracts twice within spherical water droplets, changing direction and reversing the order of colors we see.



G uitraviolet F,F# violet E indigo blue cyanogen blue D greenish blue C# green C yellow A# orange-red G#, A red G infrared ThephysicistHermarurvon Helmholtz deviseda correspondence between the visible spectrum and musical scale.

chapterfive), our neuralnetwork is preparedto registerless than one octaveof light, this palticular seven-step slice of the endless electromagnetic spectrum.Theiange of colorsis exactlysevenbecause theseare the only combinationspossiblewith the tiny "rods and cones"on our retina.Rodsperceive black and white but three kinds of cone, sensifive to red, green, and blue, combine their input into seven colors, like the seven areas within three Borromean Rinqs. Look at the screen of a color TV (while it's on)-with a ;agrufying glass if you have to-and you'll seea fine net of red, green, and blue holes responsible for the full seven-color spectrum we see,matching the receptorsin our eyes. In effect, coiors are not "out there" but always within us. A spectrumis a colorlessspreadofvibrations of-energyfrom which we make a Dicture. Nature is consistent in her pattems. Oddly enough, the colors of the planets past earth happen to form the familiar spectrai sequence: Mars-red, Jupiter-orange, Satum-yellow, Uranus-green, and Neptune-blue. peihaps the solar systemis a refractionof light from within the galaxy and the planets show a spectrum arranged with the harmonies of a



were not far off when they musical scale.The Pythagoreans harmor".t" .i tft" music of th"espheres, a celestial choir hear' nlizine in octaveswe cannot ordinarily wnrcn Tt"r" *ut another part of Newton's exPeriment' but wirich has profound implications He . is t"sr". Lo*t mto took a secondprism and directed the colored spectrum other ii f. f'rit amazementthe colors reemergedfrom the whole wtute side of the prism reunitedas a singleshaftol -it3"ff,l"tt light. Just a; the Monad createsall numbers yet never 'tose's as the whole string retains its tone, so it is with the light. itegardlessof appearances, sourceis never lost oI far from us.

THE RAINBOWBRIDGE and In our day of proliferating self-help, self-awareness, workshoPs,books, tapes,vldeos' soitself-development it ware, and multimedja technology, may be difficult,to oPen discussionof the rernemberthat until fairly recently inner life hasbeen taboo.For countlesscenturiesthe deePer was understanding of our irmer stuucture-function-order walls, mysteryreligionsand oral tradikept within te-mple tionsamongsmallgroups,which soughtto avojd drewrath and the misuse of their knowledge of relieious=intolerance The bv oth-ers. significanceof the number sevento our inner li'{ewas kept disguised in mystical terminology,metaphor, it and myth because refersto a great secret-the knowledge of self-transformation.



Onr, tro, three, four, fizp J-'et e'."




AII goodchildren to go heaoen. -{hildren's song

Worldwide tradition acknowledges humans as being composed in accord with natural principles and made in the A image of the cosrnos. useful metaphor of our irurer life is that of the prism and sPectrum. Imagine the existence of a divine light beyond measure/the mystedgreat, conscious, ous power of awareness itself, shining through the Triad of our Higher Self,asif through a triangular "prism. " It refracts into seven "colors," seven centers of gravity, the ancient seven-tone"scale" of our soul. And upon this inner specruIed by the powerful hormones trum our body precipitates, of sevensetsof endocrineglands. The seven centers-of which more will be discussed later-are often associatedwith areas of our body: at the baseof our spine or sacrum,our genitals,solar plexus (just below the ribcage), chest, throat, brow, and crown of our head. To distract the uninitiated the ancients disguised their knowledge of seven levels within us as myths and allegorids, in mystical terminology, symbolic art, crafts, architecfure, theater, dance, and as many of the s;.nnbolslisted at the beginning of this chapter. Self-transformation was the significanceof the mythic "rainbow bridge" of the Greek Iris, of the Norse Odiru and of Native Americans, linking earth and heaven, connecting our "lower" and "higher" natures. We are the seven-stringedlyre of Apollo, whose music is heard by both the gods and humans. The understanding of our sevenfold nature is hidden in the astrological symbolism of seven planets, the seven traditional vowels from alpha to omega,seven-headed snakes,sevenchambered caves of initiation, seven-steppedladders to heavenand the sevengatesof the "underworld" within us through which the Egyptian initiate had to pass toward liberation into light. The seven"liberal arts" of Medieval education-grammar, rhetoric, logic, arithmetic, geometry music, astronomy-were intended as such a ladder to "liberate" us from a mundane life into the greater awareness and understanding worthy of the heavenly spheres.They are symbolized by the seven metals of the alchemical processthat transmutesbase lead (our familiar nature) to divine gold (spiritual adulthood). In seafaringsocietiesour seveninner levels of experiencewere known as "the sel'en




Aztec depiction of seven "caves of initiation" within ourselves.

Lesser Rose Window of the Cathedral of St.John the Divine in New York Ciry world's largestcathedral (and still being built). Note its heptagonaldesign,one of the very few sevenfold Rose Windows in existence. Odin's Rainbow Bridge to Valhalla.

seas,"acrosswhich none but the most worthy could travel. Our sevencentersares)'rnbolizedby seven-storystructures, mor.rntains, and temples, from Chinese pagodas to the Potala, the former residence of the Dalai Lama; to the seven levels of Babylonian ziggurats, each level of which was colored to representthe sevenvisible celestiaibodies/ as were King Nebuchadnezzar's Hanging Gardens, one of the ancientworld's SevenWonders.The centerswithin us have been called the "seven daughters of Atlas," whom Zeus made into starscalled the Pleiades,the sevensisters.Lr legend our centers were the "seven children of Pythagoras," three sons and four daughters, who may or may not have been historical, but who would be interpreted by a philosophical geometer as the hiangle and square, the divine within us capping our mundane self. Our sciencesand mathematics are dominated by words having Greekroots,but we defer to India for terms describing details of our inner life, which were mapped long ago by Vedic priests. The Sanskrit term chakra("wheel") is popu-

Reconstructed Babylonian Ziggurat, model of both cosmos and self.



Babylonian priests anoint the Tree of Life, an early caduceussyrnbol.

Mesopotamian cylinder seai

Roman caduceus

larly usedto describeeachof our sevencenters because they have been said to appear to the subtle sight of r""rc u, Catherine wheels, spirming fireworks. Perhaps the most widespread and familiar of all ancient esoteric syrnbols of the Self to survive the centuries is the caduceus,a urdversal icon of medicine and healing. It was sometimesdepicted asa winged staff with two snakls twining up around it in the forrn of a double helix, crossing at eachof the five c entralchakras.ltis startling to seethis sa-me symbol depicted in so many different cultures. But its pervasivenessshouldn't surprise us since the discovery is a humanexperience, identical for all regardless the pjrticuof lar_cultural s)rmbols used to interpret and expresJ it. The caduceus was the staff of authority carried by the Ol;.'rnpian messengersHermes and Iris, and it became the medical s)'rnbol of health when understood and adopted bv Asclepius, Hippocrates,and other famous healers. The central staff of the caduceus is traditionally taken to represent our spine, upon which the seven centers are often shown attached like flowers on a stem. The path of the two spiraling snakesspansthe five middle centersfrom genital to brow, syrnbolizing divine creativeenergieswi thin is that travel the spine in a subtly spiraling *"uui. Her-us and Iris carry the caduceus as a symbol of their authority to travel everywhere between heavery earth, and the underworld. the mythic world within us. The caduceus,or sphe, with its centershas been symbolized by seven-branched candelabras and sacredtrees,bv


the world axis, world mountain, malpoles, magic wands' swords, anchors, sacredrivers along seven cities, and by the siaff of Moses,which becamea serpent and saved "all who meditation under thebo gazeupon it." Buddha'sseven-year f,ee (from the SanskritBo,Budh,wotds sign:'fying"awake") represents our irurer joumey through seven levels up the ' "tree" into enlightenment. But these seven centers in us are not mystical tales, They arefamiliarJo pagan superstition,or abstractconcepts. motiaations,thc W" krro* them asour psychological "rrJryor,e. of the passions,desires,emotions, thoughts, and ,p""i.orn intuitions we eiperience, each with its own countless the shades.The Pythagoreansca11ed number seven "the vehicle of life," alluding to thesesevenlevels of motivation within us. Like eachstring of Apollo's lyre and eachcolor of the rainbow, these centersvibrate at different rates and carry different qualities. We commonly associatethese inner centers of subtle energy with parts of our body in phraseslike

others Hr rr,o knows is but zuise, hewhoknows ened' elf hims is enliSht -Lao-tzlj

Hu*ormusic is the whichmay be harmony who knownby any person of turns to contemplation himself. . . lt is this that join togethertheparts of the thesoul,and keeps rationalpart unitedwith theirrational. -Gioseffe Zarlino (1517-1590, Italian composer and musical theorist)

Au rto, isin tune with is Thee, Unioerse, in O tune with me! -Marcus Aurelius

Alchemicalapparahs for purifying "metals"

TheHand of God,Celtic sculpture





a I


Aztec worship of the double sement

Sumerian sculpfure

The centralsecretis, therefore, know that the to oarious humanpassions andfeelings and emotions in thehumanheartarenot wrong in themseloes; only theyhaoeto becarefully controlled gioena and higherandhigher direction, until theyattain theaeryhighest condition of excellence. -Vivekananda (1863-1902, Indian religious leader and teacher)

"thinking with his groin," " gft feeling," "have a h eafi," " all choked up," and "use your head." But what we actually feel at these places are universal energies streaming through us in rising and falling waves of differing iengths, intensities, and durations. The two ends of our inner caduceus,sacrum and crowry forming its axis, act as two poles of a battery. Although their functions aren't apparent in ordinary human experience, they awaken in advanced stages of sel{-development. Notice how the two "snakes" do not reach the staff's ends but crossat five centersbetween the genital and brow. At the top their headstake the shapeof our eyebrows. Ancient twelve-tribe nations hid this inner symbolism in their astronomical and zodiacal myths. The slven visible celestial bodies listed in their traditional order (using famiiiar Roman names)-Sun, Moon, Mercury, Venus, Mars, Jupiter, Satum-were associated with the crown, brow, throat, heart, solar plexus, genital, and sacrum. The illustration associatesthese seven centers with the seven planets in *reir "home" constellations. examining their slrmbolism By we uncover a beautiful allegory describing the birth and ascension the divine power of consciousness of within us. One of the most misunderstood descriptions is flre final chapter of the New Testament, Revelaion of Saint lohn the the Divine. Written in mythopoeic symbolism, its descriptions of the transformative process startle the reader. In his




of aware *rrt be

and thewisdom the that strength is in themif is theirunderstandingto beexpanded' -Vauvenargues (1715-17 47, soldier, rnoralist) French

o7 aiycuuy Olt, tne of fixing theattention men on theworldwithinthem! Zodi.o Th6Apo6lyptic ' /.'^ Uuna -Samuel Taylor Coleridge (1772-1834,Engltsh Poet and critic)

"q*.. .u"",*.'oi -i ,r9Lrr,,,* - 5-,;"S"., in Cancer



Throat - Mer.cury in Virgo

Air Water

- Venus in Libra

Plexus - Mars in Scorpio


. Jupitet in Sagittarius

E',',"^" 5"n",*,'Q^a*^ ".



vision John describes a "book written within and on the backside,sealedwith sevenseals."We are that book. a beautiful workof nature, He describes "opening" of eachseal.The first to open the is second from the bottom, the genital. John describes it as andhelp yourself further to "a white horse: and he that sat on him had a bow . . . and he grozoth; that'sthebest went forth to conquer." He alludes to the s)'rnbol of 'Jupiter thing. in Sagittarius," the cosmic creating force, as "hunter." \Atrhen creative energies passing through us focus at the genital -Moses Auerbach level, we become aware of sexual passion and the roving (1812-1882, Germannovelist) hunter awakens. Sex in our civilization is both adored and reviled. The promiscuous Jupiter, the Greek Zeus, is always associatedwith a middle. So a human body drawn to fit within a squarehas its genitalsat the middle. If we can pay Choose alwaysthe way attention to the force o{ this energy as it arises in us, distinthat seems right; howeoer guishing thelellzg of the energy from its effectsand the values we give them, we notice its power growing and recedrough it may be.Practice ing in strong, Iong, slow waves characteristic of this passion. wiII makeit easyand As we constructively tap our sexual energies, we acless the pleasant. power of the universal creatingprocess. The second center, associated with the solar plexus, is -attdbuted to Pythagoras describedby Saint]ohn: "And therewent out another horse that was red; and power was given to him that sat thereon to take peace from the earth, and that they should kill one another; and thete was given unto him a great sword.,, In zodiacals).'rnbolism see"Mars in Scorpio,"the red planet we Thrrc i, nomanalone, and stinging scorpion, the god of war with raised swoid. We because manis a eoery often see statues of warriors with Mars depicted on their microcosm, carries and the armor, at the gut. The effectsof energiesfocusld through our whole solar plexus are often difficult to deal with, as they have worldabout him. extreme positive and negative attributes. On one hand, you -Sir Thomas Browne know how-you feel when you're angry your gut tightens; power swells within it. In the solar plexus we experience the feel of fear and terror, Phobos and Deimos, the names of the two sons of Mars and also the planet,s two moons. Anxiety about school gives some children an upset stomaclu and ,,butterflies,; in the .pit of the The hearthasits reasons actors witlL stage fright get stomach."Have you noticed how illnessand weakinine folwhich reason knows low strong attacks of fear and outbursts of anger? Uidue nothingof. focus at the solar plexus is too often characteriZed by concem with what one can get, so desire, greed, and en\,y are -Blaise Pascal also associated with it. On the other hand, the solar piexus

Rej oice being our in y self



of is also the place to call upon for the noble characteristics courageand endurance. -- --Vrfi"t"ut the solar plexus is concemedwith what it can with what it can Loae, whois most *re net, tLt*t"'eutt, thiri "seal," is concerned is that of vmus' thg g9dsymbolism amongthe beautifuI F;;;;;;;;;;cal r.a beauty,wife of Mars and mother of Hari"s.i'i.* immortalgods,themelter home in the constellation of Libra' the Balance -""1", in of limbs,ooerwhelms who "had a ", i-J".-3"i.i l"r*'s vision describesa horseman their heartstheintelligence in his hand'" our heart has its own way of ;;;;;f b;#;"' ways transcendlng l9i' of of weighing events in and wisecounsel all il;;L;, called this "the inielligenceot the neart l,lc' The Ee\;tians godsand all men. at t rre sJmeoneor somethingyou love and feel the warmth -Hesiod ;iy""t chest.Suitain attention there' breathe with ;h;;;;; to u.a f"a ihe glowing warmth' Unless we succumb ii, too mlich, wJ can feel the beauty of the. heart iiu"t*g and know its urge to give and love unconditionLf"fa;ig a1ly. - 'Or, ,1't"other hand, the heart is all too often ihe unwillWhrnyouhauebeen -l ing container of repressedgrief and-angui:.h,*gl.?tt com?elledbY diiappointment. We cancarry heartfeltemohonsall th-rougn 'lirr"" to we may bury them Medical circumstances be even though orr. oi strong are finding that lifelong repression researchers in disturbed anYmanner, can emotionswithout healthfuI reiease lead to heart attacks' quicklyreturn to Yourself, Sustainedfocus of attention at the heart centercanbe one of out and do not continue of the most beautiful ways to experience life, but when the tunelongerthan the heart opens, those emotions,which have been most iorcenight fullv repressed,burst forth first. Some find this "dark lasts'YouwiII coffipulsion to face and shut down their emotions of tite soul" difficult control increasing harse with logic and explanation, requiring a cleansing flood of o.)efyouf own harmonyby tearsto finally releasethe tension. felt continuallyleturning to itAmong our sPectrum of seven centers, motivations at the gerJtal, grt, utta heart are most famiiiar to people' -Marcus Aurelius TogethEr with hore rarefied motivations from the throat u.,d bto- centersthey weave the Web of Athena, our Personality pattem, which the Greekspersonified as the goddessPsyche,our divine soul. Plaio wrote that io attend to the soul's development is the highest good we can do. The ancientswere concemed with the soui's purification and transformation As they saw called "desire" inevitably passes it, the stream of "ttutgy cannot and should not try to cease through us like a river. We



Thrrc orcohurdred and onearteries thehmrt, of oneof thempenetrates the croutnof the head;mooing upwardsby it a man reaches the .immortal. -Khandogya Upanishad

Use thetight thatis in youto recooer your natural clearnesssight. of -Lao-ilzu

His aim shouldbeto

its flow, but we areresponsible its qualities. example, for For are we set on mindlessly acquiring more and more "things," or are our desires oriented toward wisdom and understanding? Do you seethe difference? The ancients wele concerned with such purification of eachlevel by facing, releasing, and reorienting the energies of each center toward finer and more noble expression. Experience with the centets above the heart involve advancedstagesof meditation taught by a qualified instructor. Purified creative energies brought through the throat, brow, and crown call forth the opening of the "seventh seal,' at our sacrum.The namesacrum, from the Latin for "sacred.,, is a curious name for the triangular bone embedded within the pelvis. The Freemasons symbolized the sacrum and spine as an upright shovel with which we dig the foundation of our "temple," our transformed self. It is the most profound and mysterious of the centers,and its lawful opening releasesa kansformative energy that results in a metarnorphosis beyond what it is to be human. It is variousiy referred to as the crowning of the sun god, the arrival of the conqueror, and the return of the rnessiah, ot avatat crowned with seven rays and surrounded by twelve stars. We miss the point if we misinterpret the ancient myths by literalizing this divine emergenceasonly having once occurred to someone else, outside our own being. The ancients taught that it is zrewho have this possibility of liberation. - Continuing the metaphor of the prism and the spectrum, when a1l seven centers have been purified, their colors recombineinto whole white sunlight and are offeredto their source "above" or deep within us. Purification is a maior

concentrate simplifu and , andsoto expand being his . . . andsotofloat upwards towaldsthediaine of whose fountain being stream flowswithinhim. -Plotinus



concern of fairy tales. It was only after Snow White cleaned and organizedthe home of the sevendwarfs that they were able tJcome to her rescue,allowing the prince to awaken her with his kiss and then take her to live in his father's divine castle. The details of this processwere the most closely guarded secretsof the ancient world's religions and mystery schools' we Like people of all erasand places, can discoverthe archetvpai source of our motivations for oneself only by direct experience.And we must start exactly where we are, simply by becoming more aware of our motivations and by perceiving our inner life as dynamic waves of passing energy. In ancient times, becoming aware of oneself as an energy system was symbolized as finding one's way onto the rainbow bridge linking"earth" wlth"heaven." Life was likened to the song played on Apollo's lyre; one tuned and ascended this musiiil scale of7 Apollo's lyre within oneself. The ancients sought to ttrne or purify themselves so that the unseen divinity deep within them could bettel Play uPon their instrument in the world. To the ancients, the task of constructing the universe meant reconstructing themselves. In democratic societies of today we are not constrainedas in the past. Timelessmodels of ourselves and tools for self-awareness are becoming understood again. It is uffiecessary to hide this knowledge of our inner constitution. It is more imPortant to understand the sources of our motivations and to transform them' Once we catch on, we travel in seven-leagueboots.

Native American mask, Siatue of Liberty, and Egyptian Goddess Seshat ("Seven") each represent an enlightened being crowned with seven rays.



Each of us is a tripart whole, an individualized field of interpenetrating light, energy, and living matter, or Higher Self, psychg and body.

The seven-stagetransformation of our elusive inner life, the goal of ancient seers,philosophers and initiates around the world, does not graft miraculous powers onto our familiar Self; it changes us from our core. After all, a butterfly is more than just a caterpillar with wings. The process transforms our very mode of pelception: we see ourselves directly, not through reflected pictures of ourselves and the world but as the vely energy that motivates those pictures. It's no wonder that the sevenlold nrocessesof nature and mathematicshad to be made metaphoricalthrough mythology and religiory fairy tales, and architecture. Seven is an unbom virgin, etemally elusive when chased by words or the geometer'stools. But, to our surprise,it is the Heptad's role in our inner awarenessthat is also doing the chasing.

Untwisting aIIthe chains that tie thehidden soulof harmony. -John Milton (160&-1674, English poet)

IHnptrn Irnttt

Rene Periodic


Change has an absolute limit: This produces two modes; The two modes produce four forms, The four forms produce eight trigrams; The eight trigrams determine fortune and misfortune. -Confucius (commentary theI Ching) on Now this, monks, is the noble truth of the way that leads to the cessationof sorrow: this is ihe noble Eightfold Way; namely, right views, dght intention, right speech,right action, right livelihood, right effort, right mindfuLness, right concentration. -Buddhn Mathematics takes us still further frorn what is human into the region of absolute necessity,to which not only the actual world, but every possible world, must conform. -B ertrcnd Russell 1872-1970. British mnthematicin ( t andphilosopher) ln the uncertain hour before the moming Near the ending of interminable night At the recurrent end of the unending. -7. S. Eliot (1888-1955, Bitish poetandcritic) You can't fold a pieceof paper in half eight times. -Malh folklore

The Birth of the Octagon. 267

BEYOND TfIE EIOHTBALL Passing beyond elusive virgin number seven, we lbap to eight and find ourselves in a quite different place within the Decad. Eight is promiscuous, having more divisors among the Decad than any other number; it is divisible by one, two, and four (8 = 1 x 2 x 4). Its divisors tell us that its roots are clearly in the Monad, Dyad, and Tetrad. The ancients called eight "justice" and "evenly even" because it can be continually halved all the way to unity (unlike an "oddly even" number like six, which cannot reachunity by halving). The Indo-Europeannamesfor "eight" emphasizeit as a "doubling of tour," most deriving from the Sarskrit o-catasrah ("twice four"), which became okta in Greek and oclo in Latin. These referencesto one, two, and four tell us that the archetype of eight called Octad, weaves together the principles of Monad's unity, expansion, and cycles, Dyad's polarity, and Tetrad's materializatior! that is, material forms precipitate from their archetypes through polarity and pulsing cycles. Within the Decad only one and eight are composed of threeidenticaldivisors(1 = 1 x 1 x l and 8 =2x2x2),u'td so ihey projectinto three-dimensional spaceascubeshaving sidesof one and two, with volumes of one and eight. Every cube has eight corners as if to emphasize the mutual relationship between the number eight and this form. The cube's dual, the octahedron, has eight faces dividing space equally in eight directions. Square faces link the cube and the Octad with the Tetrad's syrnbolism as " earth," manifest form, the crystalline phase of rnatter, and physical stability. This is why the nurnber eigh! like the number four, is often associatedwith the Great Mother Goddessin her nourishing aspect. FoIk referencesto "eight" are not particularly conunon. We speak of the "figure eight" in ice skating and of being "behind the eight-ball." Other than the word "eighL,', what's their connection? The figure eight represents a polar cycle peryetuaily swinging between extremes and crossing itself in the middle as the diagonals of a square. Turned sideways, the figure eight is the mathematician's glyph {or the infinite.

aaao aaaa

rx6 t-l

aa ao o o o a

aa aa

o a

a a

L ^ t' * 8x1

aaaa aaaa

One and eight are the volumes of the only two cubes within the Decad.



The significance of the phrase "behind the eight ball" seemsmoie obscure unless we realize that the black billiard ball numbered "eighf is like the crossing point in the figure eieht. It is the "oddbali" at the exact middle of the fifteen with seven lower-number solid-color balls below and ba"1ls, seven striped balls with higher numbers above--To be "behind th; eight ball" is to {ind oneself in the middle of a dilemma with {seven) elusive solutions on each side' Spanish gold coins known as The seventeenth-century "rcaIes," and favored by pirates, were called "pieces ol eiqht" because each coin was perforated into eight pieshiped sections and could be broken into fractions of their fulivalue. TWo "pieces of eight," one quartel of the coin, is the sourceof the use of the American slang term "two bits" to signi$r a "quarter" dollar' 'num6er eight is particularly emphasized in the The mythology and religious symbolism of the Orient. The Chitt"se t"rrere the Eight Immortal saints, eight s)'rnbols of a scholar, and eight directions of the wind. Buddhisrn is known by eight auspicious emblems and particularly by the Eightfold Path leading to enlightenment. To understand the significance of the Octad in culfural expressions we must in examineits appearance geometry nature,and art.

Notice how the octagonal shape is easily identified from a distance.

{J ""o=' I am Onethat transforms into Two, I am Twothat transforms into Four, I am Four that transforms into Eight. After this I am Oneagain. T

my4h,who gainedtheir stature TheEight Immortalsof Chinese wish us good luck. by studying nature'ssecrets,

-Eg)?tian (Hermopolitan) ara:rinn mrr+h


THEBIRTI.{ OFTHE OC]aGON The geometric construction of the regular octagon calls upon eight's divisors one, two, and four, numbers of the Monad, Dyad, and Tetrad. This tells us that the construction involves the circle, ztesica piscis, and square.The way their principles cometogetheras the Octad is best understoodby facilitating its geometric birth through tiner:esica piscisas an octagon. (1) First, construct aoesicapiscis and a squarewithin it. This squareis the construction's "seed" or-guiding form. (2-5) With the compassopen to the radius of tfie inner circle, put its point on eachof the square'sfour cornersand turn a circle centered at each comel (6) Using the straightedge, extend the original square'sfour sidesuntil they completely crossthe four circlescreatedat the corners.(7) Connect ttre eight points where the extended lines intersect the circumferences of the two large circles to (8) make an octagon. (9) Use the straightedge to connect every other cornei oI ti-Le octagonto createtwo interlacedsquareswithin it. (10)Also extend eachof the octagon'ssidesto form interlacedsquares around it. The octagonand interlaced,or double, squareare the basic structures necessaryfor exarnining the Octad,s principles.



EXPLORE THE OCIAGON'S GEOMETRY Use octagons you construct as starting frames to explore the form's intemal structures and pattems. Using the sfraightedge, connect comers and secondary crossing Points to reproduce some of the pattems shown, or your own. You may wish to make the constru ctronsaudibleby tsing wood, nails, and a guitar string, piano wire, or nylon fishing line to configure the lines of the construction. Plucking the "line," or string, between differmt points will enable you to hear the octagon's intemal proportions in relation to one another. You will discover some tones that are identical with



H<[Lf ffiffi '-v\ \ r /z \'| t/ \lav,





@ffi&s others but higher or lower in pitch. The octagon's geometry is sirnilar to that of the square, built on the "root-two ratio,' (1..414.. to 1), which you will hear. . Octagons alone will not tessellate,or fill flat space,without leaving gaps. But they will do so when combined with small squares. constructthis familiar pattem seenin tiles, To wallpaper, fabric, and elsewhere,construct or doodle an array of circles aligned in square-pattem close-packing (as opposed to hexagonal close-packing). Fill the gaps with small, tilted squares, connect their corners, and watch the circlestransform into octagons.



DISCOVERTHE OCTAD'S PATTERN IN LIVINGFORMS We've been seeingthe most widely varied geometriesoccur in the living structures of the sea. Along with circles, spheres, triangles, squares, pentagons, hexagons, ,and the five Platonic volumes, the geometry of the Octad can be {ound in the structures of such creatures as jellyfishes and "glass" sponges.Placetracing PaPerover each image, and coro,ect ih" iomers and crossing points of the interlaced squaresand octagonsthat encloseeachform to discover the creatures'underlying geometries'

Seenfrom below, many jellyfishes and "glass" sponges display octagonal strucfure.

Male maple flower



THEBREATH OFTHE COMPASSIONATE Islamic tradition holds that there are one hundred namesof God but only ninety-nine are knowable and speakable, and they are called the 99 Beautiful Names of A1lah. The highest pronounceable name is "The Compassionate." From its infinite goodness the Compassionate exhales and inhales. Through the polar cycleof divine breath the universeis periodicallycreaied, maintained, dissolved, and renewed. Islamic design, in which the depiction of animals or humans is forbidden, symbolizes this process through geometry. The pattem known as the "Breath of the Compassionate" appears throughout Islamic art and architecture, in wa1ltiles, ornamentation,doors, rugs, manGod created unioerse uscript illumination, and elsewhere. lnscreens, the constructing this throughtheBreath the of pattem the geometer calls upon the three factors of-eight (one, two, and four), weaving the principles of the Monid, Compassionate . Dyad, and Tetrad, expressingcycle, polarity, and material -Mohammed (c. 57M32) rorm. Begin with a central point through which the unknowable emerges.Turn a circle, dupli cateit as a oesicapiscis, al:rd, (1) construct a square (symbol of material form in the four states of mater). The archetypes are becoming visible through the creating process. (2) The square e,iolves to become an octagon and interlaced squares containing both elements of the fulI breath. (3) VVhenthe eieht th. "o.nerJof interlacedsquaresare unfolded, p otnttngitward,we create the symbol of cosmic expansiory onJ exhalation of the Breath of the Compassionate.(4) A second construction tums the corrers inward to depict contraction and inhala_ tion. (5) These"breaths" will "t-essellate,', fill the plane ir or all directions, leaving no gaps, just as Divine Compassion flus tne uruversewithout gaps, creahinga geometric motif seenthroughout Islamic design asa remindei of divine pres_ ence.This motif tells us that the processbv which the aiche_ typal pattems crystallize into material ctnfigurations is a polar processas simple and ephemeralas brea-thing. And so it is in the basicpositive and negative chargesof tle atoms that configure the universal patterns. We ca; share the com_ passion that createsand fills the cosmosby feeling it along with eachof our own breaths.

r""Form as polarized cycles Expansion The complete Breath of the Compassiorate





Islamic pattem

Wet llloter

Aristotle's depiction of the four "elements" and the qualities that form them.

of the More than iust an omamental motif, the Breath is a cosmological model symbolizin-g C..o"tti.""tJ -the of polarities that manifest form lslamic tradrhon' intemlav *hoie orieioutors voraciously hanslated all the ancient and medical textsthey could mathematical, Greekscienltific, f,tut, on one level, its eightfold pattem refers to fi"a,l"fft ot "t or Aristotle's diagram of the four elements, modern states = gas' fue matir \eatrh = solid, water = liquid' 2i1 -u*" ". = electronic plasma), crossing with,four. qualities (cold = wet = dissolution' dry = cryscontraction,]iot = exPansion, tallization)- On this ;ightfold loorn the archetypes,precipitate. Each geometric irihalation and exhalation of the Comthis eightfold weave of the configured passiottate"represents cosmos. Aristotle leamed this symbolism from his teacher Plato' who had studied the Pythagorean teachings Pythagoras had learned it in Egypt, where it took the form ot the Drimeval oedoad of Heimopolis, the four pairs of unmaniresponsiblefor giving birth to the ?estgods aid goddesses manifest universe. Eishtfold qeometry is an ancient symbol for the Great tv't.rttre"r Godd"essas nourisher. Her substance clothes the archetlpal pattems by crossingthe four elementswith four (two ptirs ;f ) oPposite"qualities." In this role shehas often been'depicted mythologically as an eight-legged spider


Peruvian gold nose ring' Half a circle contdirs four spiders'



(from the word "spinner"). In Native American rnythology, for example, the Grandmother Spider weaves the web of matter. In modem terms this would be done by intricately crossing polarities, electrons and protons, the warp and which litwoof of eachatomic knot. The Gre ekw ord Kosmos, sigrrifies "embroidery," and many of the great goderally desses of myth (the Egyptian Neith, Nutet and Isis, the Greek.Athena and the Three Fates, and the Mayan Ixchel) are spinstresses, weavers, and embroiderers, manifesting the world while spinning the threads of fate for humans and the entire cosmos. Spiders are not insects,although both are "arthropods" (frorn the Greek for "iointed feet"). Insectshave three body sections, six legs, and two antennae,while spiders have two sections,eight legs, and no antennae. Spiders are arachnids, named for the Greek maiden Arachne, who impudently

Native American carved she1l depicting GrarLdmother Spider.

how The spider's touch, exquisitely fine! thread, and Eeels each at lioesalongtheline. -Alexander Plate from Panarna


Eight legs of the spider, eight tentacles of the octopus, and eight directions of a Hindu Kali Yanha meditation mandala all s).nnbolizethe ensnaring, devouring, terrible aspect of the Great Mother Goddess.



challenged the goddess Athena, inventor of weaving, to-an embroiiery conjtest and was turned into a spider for her alrogance. Iir her terrible aspect, the spider goddess is the ensnaring feminine dangiing from the heavens to hap the unwary in"her web. Foreboding doom, she ultimately provides her catch with a late ol spirifual transformation and renewal' Numerous spider ernblemshave been for:nd throughout lhe world, from-Crete to Germany, Mexico, and Peru. The eightfold terrible transformer has also been represented as an octopus with its eight ensnaring arms, which spiral to reinforce it as a s)'rnbol of transformation'

THE PERIODICTABLE OF THE ELEMENTS Scientists have found a modem analogy for the eightfold weave of the cosmic spider web in the Periodic Table of Elements, the "rllLap" ol all ninety-two naturally occurring atoms from hydrogen to uranium. hr the 1860s,when chemists began grouping the known elements according to similar physical properties, they disA coveredthat the properties,repeatin periodic cycles. table energed of vertical columns called "families," which have similar properties, and horizontal rows marking gradations


II & Mg C!

Itr IV V VI \'II Tiansition elements ilIVVVIVtr Sc B N


i N!



VIII Co Ru Rh Rc

I Ni

II cd Hg ln



s o S.

K Rb Ca

Ti Zt Nb

Cr M n

Cu Z^ G. Sn Pb

8r Kr I X!

Sr B. R! L.

Pd R


Tc Po









Pr Nd Pm Sm Pr

Gd Crn 8t



Yb Lu Md No

Ni Pu

cf Es Fm

PeriodicTableof Elements



of those properties, each element having one more electron than the element to its left, thus forming ,,spectrums,,of matter acrossthe table. Eachelementhas properties similar to those directly above and below it, a litile different from those to the left and right. The chemists discovered eieht main groups, or ty?es, of elements. Scienfists now theorize that atoms and the world,s atomic configurations are not ,,things,' but whiripools of energy.,.swirling clouds of negatively iharged electrtns sur_ rorlding a positively charged core, or nricleus, of protons and neutrons. reason atoms,properties The the recurin perr_ odic cyclesof eight, as the scientistseventually discovired, is that elementsin the samecolumn have theslame number of dt:.!:o:t,b:y"y one and eight, in the outermostperiphery ot their.orbits. The "group,, number at the top oi eu"'h.oi_ umn tells the number of outer electrons. \A4renchemical ele_ ments combine *rey form molecular architectures deter_ mined by the numerical and geometric patterns of the outermostelectronrings,or,,shells.,, Atoms stuiveto com_ bine with other elementsthat will give therr-eight electrons in their outermost orbit, making tfiem stable ird satisfied elements with a "full shell.,,Atoms the extreme at rieht col_ ? l ' l f umn of the table, known as the inert, or noble, gales like h.'sr/ .--r neon and argory already have a full outer shell of e"ishtelec_ trons and are complete unto themselves. They are r:eluctant OraAztt to combine with other elements. lrVhenthe ouier shell is full row or "period,', of elementsbegins again with one .,' --+- 'e electron.Eachconsecutive atom has sirrJlu, pfryri"ut."t ical, electronic,optical, or crystalline propera"s similar "-_ io ? t\ . , ? thoseabove. "Fr*l\\ Fgr consider the elementsoxygen and sulfur, .. 9x1mple, ---*-/ directly below it. Both have six electronsiritf,eir o"tu, ,t Ltj Sq44.Q si{l cnemicatproperries.They will eachgladly com_ 31-{ with,two atoms of hydrogen, whose single electrons :T: complete wiu the outer shell with eight electrons.O"yqen and hydrogen combine to form H2dor *ut"r; *tfriu'.,J hydrogen form HrS, or hydrogen suifide. eight sulfur atoms by themselves,in fact, wilj share their outer electrons to


?\',? h's/ ?/"re 6r ,--G-q


F / 'o- \ E'\


ft ? ( ' , t 9 f \\-sl/ \'--s_d '---'

/f r--,\\ 4vt*

rurI sne.u elght electrons. ot Modem chemistry is discoveriirs molecular, atomic, and subatomic structureJ.t.iti"jf y ri_l


in an-octagonal (56)eucn ring atom no* huuing-l

?l/ate', 7/l6eea4/e



and architecture' ilar to the geometricpatternsof islamic art "-' rf,"'p""".Ji" Tabieof Elementsis a modern cosmologiBreath of cal model, a contemPorary depiction of the .the web of the Grandmotner Compassionateand the cosmic

;;td5;i;l;,h";eb hot Opinion says and nld, but therulitY is sPace. and atoms emPtY -Democritus (c. 460=370B-c., Greek philosoPher)

a uni;er;e ,fi" ft4"""a, Dyad' and Tetrad' That is, it depicts and negative charges9j -:r:; woven of polirities, Positive ar't atomic maiter, and recurring eight-step cycles to malulest rePresents matter.Eachhorizontal row of the Periodjc Table each ut oifr"t''o"tu""," or eighth step, up the "scales " When 'hote" is reachedand the outermost row's final chemical electron shell is filled with eight electrons, the next row' ot tist "octave," begins.The entire PeriodicTable,the comPlete of known ry[es of matter, sPans seven " octaves"' the numbe, appropiiate for the full spectrum of material configurations.

of the of theworld,weaving principles

RESONANCE:THESAMEBUT DIFFRENT In a sensethe Periodic Table is like a piano keyboard' each atom a chemical"note" along an ongoing scale'The materialized forms the atoms weave are like visible music' Each eishth atomic sroup and eighth white key on the Piano comprlsesone turi of iis spiral, repeatingthe tone and chemical properties at each rung. \ y'hethermusical notes are repeatit g^tl",ui.tott" ot elements are repeating their phygical propthe restores fundamentalbasein a erf,es,each"octave'1steP processof self-renewal,a periodic recunence,a return, but lo a corresponding rung of the spiral. The idea of peliodii (ecurrenceis at the heart of "resonance," or resounding, the same note an octave apart' We honor the principle of tesot utt"" when we celebrate holidays and aaniveisaries on the same day eachyear' A simple experiment with a guitar will demonstrate how "separate" physical objectsare linked through this principle' ' -. Pluck dif{erent strings, and listen to their notes' Now a crease very small piece of paper and hang it on one of the strings. Pluck the other string. Noihing happens to the



paper. Now tune the strings to sound exactly the same note and pluck the other string again. The paper jumps as its string "resonates," or vibrates, in tune with the plucked shing. The shings are connected through an energy iink. Now press your finger at the exact center of the string without paper and pluck onehaf The riote produced is the same as the one sounded by the whole string, but it is an octave higher. The two stuings again lesonate and vibrate together and the paper jumps again. Although physically separate, the strings vibrate at the same rate and align through spaceby means of their energy link. Resonance occurs between plucked strings as they are continually halved or doubled in length, creating the same tone in higher or lower octaves. Similar musical notesacross any number of octaveswill resonatewith eachother just as the propertiesof atomsresonate with thoselisted aboveand below them in the Periodic Table. Everything in the universe has its natural vibration frequency. Just tap and listen. A1l sirnilarly vibrating objects of the material and energetic universe are potentially linked by tesonance.You may have noticed this while driving a car. As the car accelerates, pistons oscillate,or vibrate, at differits ent rates.At a certain speed,or rate of vibration, the windows may rattle. At a higher speed, the windows stop but the glove compartrnentdoor rattles.Then the sound tecedes and perhaps the door handles begin to rattle. As the car's vibratory "note" changes,those loose obiectsthat resonate with that particular "note" will absorb its energy and vibrate or rattle with it. The Octad'sprinciple of resona goesfar beyond a ratnce tling car. The discovery of resonancein oscillating electronic circuits led to the idea of "tuning" a radio receiver to different stations instead of fixing it to receive only one station. That is, the radio receiver resonatesin s)mchronization with the broadcast waves, whose motions make the paper cone of the speaker jump and sing, reminding us of the way the paper jumped from the guitar string. On the average,the natural speaking voices of males and females are one octave apari. This fact confirms what we

a J ofar asit goes,small thing maygiae analory of great things, and showthe track of knowledge. -Lucretius (c. 99-55 r.c., Rornan poet and philosopher)



already know: men and women are "the samebut different " equal without being identical. Through the p6cess of continual halving or doubling we introduce the Dyad and polarity 1o exPand the octave hfin itely.The eighth step is the return, the periodic renewaI to the source, to the Self. We arrive at a higher or iower rung of the spiral, the samebut different. The number eight itself is the result of a triple-doubling from unity, the Monad: L -+2 -+ 4 + 8. We know that our body grows from the time sPerm and egg unite to form a whole cel1,which doubles to become two cells, then four, eight, and so on to form tissue,organs,and systelg' fut !h9 doubling from one cell to two in the processof "mitosis" (from the Greek mitos signifying " thread," referring to the of threadlike apPearance DNA chromosomesin the cell's as stages, if along the nucleus)is Jprocessof eight seamless musical scalethat renews itself upon tones of a seven-note ihe eighth step,where it becomestwo. reaching we \rVherever see an ongoing cyclic process involving polarity or doubling and resulting in material manifestation, we can expect an eightnessand the principles of the Octad. There is a curious mathematical relationship between the numbers seven and eight, the musical scale and octave, and the processof doubling. Use a calculatorto divide 1 by - .) 49 (=7x7),thatis,1/49.The answer(.0204081'63264128256. process of doubling. is an endless decimal built on the

@@@@@ division. One cell Ce11 becomestwo along an octave of eight stagesin the process of mitosis(Greek for "thread formation").


It startswith 02,then 04,then 08,then].6,32,64,128, . . .The . mystery of the creating processjumps out at us wherever we Iook. We seethis occur at the shore in the breaking of a foaming spiral wave. Incoming oceanwaves will break when the distance to the bottom below the wave eouals exactlv half the surface distance between two wavis, their "wavelength" (seechapterfive). A simple doubling relationship occurs within the structure of the human body. Your thumb, wrist, neck, and waist form a seriesof doubling "octaves."Wrap a length of string around each to verify that: * twice around the base of your thumb = once around your wrist * twice around wrist = once around your neck * twice around neck = once alound your waist Few of us will show precise results, but with many people the average relationship wiil be c1ose. Resonanceexplains why we can sometimes seethrough a "sohd" object like glass. Think about it. How many solid objects can you actually see through? I,l/hen light, either coming directly from a source or reflected from an obiect, strikes the surfaceof glass,its moleculesof silicon diodde [SiO2] absorb the energy and oscillate with it, ringing like microscopic bells, vibrating in the samerange as visible Iight colors. The tune is perfect, and nearby "molecular bells" absorb the energy through resonanceand ring with the same note. Eventually, the vibrations reach the other side and emerge as light nearly identical with their entry vibrations. Thus we "see" the object on the other side. In actuality we areseeinglight that hasresonatedthrough matter.As E=mc2 tel1sus, energy (E) is the link between matter or mass (m) and light (c). We use the word "transparent" to describe obiects like glass whose molecules resonate with light completely. \Atrhen only sorzeof the molecules in a substanceresonate or are "tuned" slightly differently, only some of the light resonates through and we see,"as through a glass darkly," that



Thorc in whom there is no guile do not rcmainin of theseaenth, place the to rest,but arepromoted theheritage thediztine of benficence whichis the etgffin graae. -Saint Clement of Alexandria (c. 150-215) 1

the material is "translucent." In/hen noneol the tnolecules in an object "ratlIe" in tune with the light, the objeci is " ooaque-" in R6sonance the world has long been recognized'Formalized in the Middle Ages asthe Hermetic Doctrine of Corand the Law of Analogy of Simiiars,it is still respondences the basis of primitive "sympathetic magic" around the world. Resonanceis behind the symbolism o{ the name of the Greek god of harmony "Apollo," which signifies "moving together." It was also at the heart of ancient therapeutic music, wherein the tones of the tuned lyre and pure voice struck chords corresponding to the tones of the elements of the psyche and body. We experience resonanceon a very personal level in our rapport with the people we know and meet. Consider that our inner world is an energysystem.Eachurge, desire/emotion, thought, and intuition representsa different wave band of subtle energy.Each of us unconsciouslybroadcaststhe energy of our inner life and receivesonly that with which we are in tune. Have you ever been in a crowd of people and felt your mood swing drastically? We resonate with the prevailing subtle "music" if we have cultivated that "tone," or qualifi within ourselves. The ancients knew that unless we purify and tune our irmer life and center ourselves at ashigh a level as we can, we wiii resonate with whatever passions are blowing in the wind. This is the root of much unconscious suffering, not knowing the sources that motivate and shape the pliable human psyche. Our outer experiences only resonate with our irrner rattle. If we don't like the connections we've formed we cannot break them, but we can transform ourselves so as not to resonate with them. It's useless to blame anyone or anlthing outside ourselves. We carry our irmer state with us and wiII find the same t'?es of friends and relationships wherever we go. To changewhat we encounter in the world we change the levels and qualities to which we attune. In myth,legend, and religious symbolism, the leap to an eighth step is traditional$ associated with spiritual elevation and deliverance, or salvation. Thus, we often see in



architecture a square base below an octagon supporting a spherical dome, the octagon syrnbolizing the transition between the square and sphere, earth and heaven. In the alphanumeric system of the early Christians, based on the Greek language in which the New Testament was writterL the letters of the narne of Jesus, 'I1oor-rs, add up to 888, emphasizing the transcendent s),rnbolism of the octave.

HARMONYOFTfIE COSMICBREATH In our exarnination of expressions of the Octad we find the ancient Chinese oracular system known in the West as the I Ching, or Bookof Changes, Regarded by modem scienceas superstitiory it is actually an interesting cosmological model. Ir:rmany ways the.l Ching is like the Periodic Table of Elements; both systems are based on various combinations of polarity and periodic renewal. But instead of looking at "things" and structures like "atoms" and "molecules,, the I Ching depic1sthe cosmos as dynamic events, as distinct



that lin-k' ."things'" stasesof transformation, the processes of natural law r ne processwas called the Tao,the way Thi"s to exPlam alL I Chinp was used by ancient Chinese sages ald aruorocesiesof growth and transformation' trom Plant t-"irri" the pro.u"i"? *" the movementsof the heavensand ** cedure oi events in our lives' Absolute' rrr-C}ri.""" myth and legend the unknowable rJ Cr"ti the "Grea1Exfteme" or "Grand Ultimate" f..o*. "t "ridgepole," the world axis),issuesthe Monacl' Tai ffit".tffu


rhis bvacircle' one'thm

th.e.ya,nf oroduces the compiementary powers of the Dyad' 'und I18ntuin, polarities of the world: Positive-negahve' acflvedark] maie-female, odd-even, advance-wilhclraw rnvolvereceptive, proton-electron, aggressive-yieldrng/ (se -"r,i-detu"ilmet t, rnotion-tranquillity, and so on its circular formieacLr-part contains a small il;;-;;;I" to endoieie of its opposite within itselJ,causing dre parts yang as a solid [lrr" .-rt"t" Ji"rt other. If we s]'rnbolize the line and vir as a broken line, their combinationstaken two "triat a timJ qive us exactly four possibilities' Taken as grams," tfiee at a time, we find exactly eiSht Possiblecomfii"urio"t oI yang and yil,. In the Processof transformation'






rI I I


U Yang and Yin takerLtwo at a time produce four Possibilities. Taken three at a time they yield eight Possibiiities. II

2 rI I I II II I I - I I I I I I








eachof the three levels comprising the trigrams can be either positive or negative, yang or yin' Each level is capable of change, and as it progresses through its eight transitional state; the " charge"of eachposition altemates.Look closely at the subtle transformative stages of the lines as they proceed from three solid to three broken lines, With your eyes you can feel their rocking rhythm. Modern mathematicians will recognize this logically expanding pattem asthe binary, or "base two," sequence counting from zero to severy the code with which modem comPuter software is writter! suited to electricity'sonly two possibilitiesof being "on" or "off." (Computer hackersmeasurea computer's memory in terms of these eight combinations as eight "bits" to the By "byte," which build to kilobytes and megabytes). examining the occurrences and combinations of active and passive forces in their various positions and relationships within each "trlgram," the sagesrealized how the rhythms of polarity give birth to all possibilities. According to tradition, ln 2825 B.c. the legendary first ruler of China, Fu Hsi" came acrossa tortoise shell. ln a flash of insight upon examining its pattem, he understood the various ways in which the cosmic polarities interplay in any situation. He composed his findings as the I Ching, whic}l was modified seventeencenturies later by King Wen and six centuries after that by Confucius. The text that has come down to us is the result of their combined understanding and commentary. If we look at a tortoise shell we find tessellatinghexagons arranged in a six-around-one pattern. The principles of the hexagonal shape tell us that structure, strength and material efficiency are at work-an obvious advantage for a tortoise shell. But on further examination if we take two hexagons as the center we see that they are surrounded by ergftfother hexagons.Fu Hsi saw in the two central hexagons the o. pposite principles of yang and yin. In the wrinkles within the surrounding hexagons he saw their eight possible "trigrams." The diagram of the eight trigrams around the central symbol of yin-yang is known as the Pa Kua ("eight diagrams"). There are actually two traditional arrangements of the eight around the center. The older, Fu Hsi's original (depicted here), is known as the Former



and is said to depict the ideai patteT:f Heaven Sequence cont U nceasingly emPlate transformations of the cosmos. Another arrangement of the by of thegeneration all things same eight known as the Later Heaven Sequenceis used earth, enabling s"omun".e., to interpret transformationson 'the and through change, harmonious placement and alignih"* to determine twself to the accustom ment fol cities, towns, tempies, homes, and tombs in relation thoughtthnt thenatureof to the landscapeand its terrestrialenergies' aboae delights Notice how eachsolid and broken line forms a symmetheunitserse try with its corresponding place across the central yin-yang all in chnngingthe things that existand makingnew stnnbol. Fu Hsi intemreted the different combinations and placeof ones thesame Pattern' ments of yang andyin as a cycle of the stagesin any Process' that For eaerything exists fhe rycle begins with the three yang lines at the top, representing "heaven," or the most active state. The process Proof is theseed thnt which und the cirdlebut in a "figceedsihrough eight stages,not a,'o shallcome "crosti"g f'rornit. ,tt" center between stages numbered ure eight," -Marcus Aurelius three ird four, where it continues around and reaches the bottom, the three yln lines representing the "earth," the most passive and recePtivestate. Being a continuous Processit ito.rus lrp the center again and returns to the most active stage of the cycle, p:ureyang. The eight stagesare an exPression of the archetlpal Octad, the result of the play of principles of Monad, Dyid, and Tetrad. The ancient names of the t igtu-t and their s)''rnbolism are said to reveal the "ight esientiaGtructure of every cycle found in nature. Read them in sequencefrom 0 to 7 and then 7 to 0. Try to sensethe flow of their cycle.



(-n ten

rui , r-: .a'.t


\. \,



,,,lif(')ill"*,." I , :U )-'ll Chensl-IIJ"l{en ! - - Y

I{ un

u | -

Ln Ien lrlr

Heaven, creative,origin, immobile, stron& firm Attraction, achievement, joy, satisfaction Awareness, separatiory beauty

^t r . r,4 -


^ .


Action, movement, development Following, penetration Danger, peril Stop,rest Earth, receptive,docile, flexible




K a n

o == / II

r\en K'iih

To examine the cosmic transfotmations in greater detail, the sages combined the eight trigrams in pairs, one over another forming six lines called "hexagrams" (not to be confused with the six-pointed star). The resulting sixty-four combinations reveal the continuous stagesin the cosmic creating process flowing from all yang to all yin and back around. They are traditionally arranged as an 8 x 8 square and also as a continuous circle. Each hexagrarn represents a fiiadl. yang, yin, and the harmonious balance between them.


The sixty-four trarsformations can be arrangedas a heavenly circle, or square eartf't.


Penetration Rest

"Gradual Progress" I I I I

Rest Penetration




The sixty-four hexagrams form the archetypal evolution of events within the continuous flow of all possibiiities. As an example, the hexagram symbolizing "gradual progress//is made of the trigram for "rcst," below the trigram for "penetation." The relative positions of the trigrams modify their significance.Inverted, "rest" caPPing "penetration" indicates "decay." The ancient sages referred to the pulsing rhythm of polarities within eachhexagram as the "harmony of the cosmic breath." Like each"exploded" and "imploded" octagon the of the Islamic "Breath of the Compassionate" hexagrams reveal the Octad's cycle of polar possibilities. IiVhile the name I Ching is usually translated as the Book the of Changes, word "I" canbe taken not only as "change" but as " easy," referring to the natural path along which the universe flows and which the sageknows as the path of ihe cosmic creating process. At any given instant the universe The by may be characterized one of thesesixty-four stages. yin and yang arepolar opposites,and eachmoment in time


is the third element, the tiad, in which they rnerge and harmonize, for each moment is harmonious within a great order.The act of tossingand observingthreecoins,threi tortoise shells, or the groupings of forty-nine yarrow stalks representsthe yln and yang possibilities of the moment they are tossed.Eachhexagramrevealsthe cosmicpolarities atwork, a "picture" of the moment. Thus the moment, the hexagram, and the question in mind at the time of tossing are synchro. nous. h ancient times no interpretive book like the I Chlng was necessary becausethe sagescould decipher the hexagrams by examining the lines and their positions directly. By interpreting each moment's hexagram the sagescould take the proper course of action warranted by that moment. The I Ching made the oracle more widely available. In their cosmological models all cultures recogrrize the interplay of polarities as the most fundamental aspect of the creating process.This is true in nature as well as in the s).'rnbolic constructions of the geometer. Hindu Ayurvedic medicine recognizessix basic "tastes": sweet, sour, bitter, salty, pungent, and astringent.Combinations of their presence or absencein food form sixty-four possible tastes we can recognize. Modem scienceis finding this arrangement in fields as diverse as computer design and biotechnology, where investigations into the genetic code have revealed DNA to be composed of sixty-four six-part "codons,', or genetic "wotds." The cosmic structure-function-order remains the same through time's transformations, whether interpreted bv ancientsymbology or modern terminology.

Tn tclringaoesnot and ffir itselfwith proofs results; does aaunt it not itself, nor is it easyto appfoach. a part of Like nnture,it waits until it is discoz.tered. --{arl JurLg

P: Phosphate C: Cytosine G: Guanine A: Adenine T: Thymine

DNA molecule

TfIE GEOMETRYOF CHESS \Ahen $e world's sixty-four transformative stages represented,by the "hexagrams" of the I Ching were a.ianged as an 8 x 8 square you may have noticed their resemblanie to a chessboard.The ancient game of chess is a cosmological model in the form of a board game in which the oppoilng forces of the universe are brought to a more personai levei The chessboard, the spider's web, the Breath of the Compassionate, and the I Ching eachin its own way represents

292 Thirteenth-centurY SPanish painting of a Christian and a Muslim playing chess.


'tlethe world's opposite forces weaving the - eightfold ments" and qt ilities. The board's sixty-four alternating black-white (or red-white) squares rePresent the world's polarities, the warp and weft of matter's web, the compasiionate cosmic exhale-inhale, the yang-yin intetplay in sixtyfour possible stages of transformatioru the conflicting is Th, ,h"r, board the choices that life Presentsus. Ttorld,the pieces the arc The game of chess was consciously designed to'repreof phenomena lhe unizterse, sent our"world of transformations on a restricted field of are therulesof theSame action. The board represents a Monad, the universe, a planet, a community, I person, a whole on any level' The whatwe call thelawsof eame's pieces reptlse.,i its elements and their possible Nature. Theplayer on the X"tions in the wotjd. The rules of chessare the ru1esof life, othersideis hidden from including those governing our own inner nature' \l/hen the us. Weknow that his PlaY Caliph of Baghdad was asked, "\ [hat is chess?" he ans*ered, "\ trhatis life?" Indeed, chesshas been called the isfair, just, andpatient. "royal game of life." Chess rePresents the macrocosm and ' But alsowe lotow,to our microcJsm, the interactive oppbsing forces and principies of he cost, neoer the universe and their correspondence to humankind's or a ooerlooks mistake, inner life and all its conflicting forces and possibilities' the makes smallest Understood in its original sensethe game provides us with a way to discuss our own transformations. an anc allow efor ignor ce. Ii is beiieved that the game of chesswas created around -Thomas HenrY HuxIeY the sixth century either in lndia or Persia but, as pieces dating from the second century have been found in Asia, its oriboard corgii may be much older. The 8 x 8 geometry.of _the the foundation geometry of-a Hindu temple iesponhs with dedicated to the divinity in the form of Shiva, the trans-


former, The interplay among chesspiecesrepresentsthe cosmos and our lives as fields of hansformation. In'Zoroastrian Persia chess played out the struggle for the world between good and evil, light and dark, understanding and ignorance, as symbolized by the god or principle Ahura Mazda and his shadow twin Ahriman. Some Persian terms have been retained, such as "checkmate," which derived from shahmat, signifying "the king is dead." The chess pieces represent forces of nature around us and psychological motivations within us; relationships among pieces on the board represent aspects of ourselves. Thus, chess gave the ancient philosophers a way to speak about irmer development in symbols and to explore various possible psychological situations represented by the garne's structure and process. Over the centuries, some pieces, including elephants and chariots, were different from the ones we know today. When chessarrived in Europe after the eleventh century, the pieces took on political and religious athibutes. Each step on the board, represented by the movement of the pieces, takes us through the polarities of the Dyad, light and shadow, the etemal struggle between consciousness and unconsciousness,wisdom and ignorance, our conshuctive aind destructive motivations. We walk among opposite forces waging battle for domination of the world and Sell

Hindu temple foundation geomeky for a temple dedicated to Shiva, "the transformer,"



enlightenment that is ensasing in the struggle for spiritual our llves '^'the"r.ilseensignificanceof eachol there (= ridebegins with sixteerr 2x 8) Piecet Wttit" "u."n ( king. TheY are another lorm possibility' representing position' power' and ;;;?g;;;t, tocationthat deterEach piecebegins the gameon a speiific minei its initial relationshiPs' "- p.t"lUifi 'apidlyt there are 20 posslU]1opel i"t "-",gu potsibilities' and 20'000posst-;;""s t""o"i-*o"" -* ;;;;t;400 The branching from their combinations' ;i: ,hitd rules' some the boird according to fixed ;#;;;;;;,*d alter;"" color throughout the game' othe$ i;;.1";; nating cJors beneaththem' "- --A'k* of each the to r.rnderstanding deepersignificance in a stralgnt rne/ etricwayit moves'whether piece is tiregeom

onr-ytyPes: -:* are six Pawn #?Ii,?il3?ifJi?li'ilt

5" ii;;#;h;

move repreht ootu.rrJ" of a triangle' or both' A " squate" algnmove rePresents .#tits eurthlv action. A triangular ment.with tire divine within us'

do",q side oi a slluare'or -diagonallv -tl"

tle Sel{' dlltl,e:f:k orr ,y''",u^, d""p"' or Highel .trl, ,ofur" ol mo-ve-

the sun The king, the most important Piece' represents


Each chessPiecemoves geometrically. The bishoP, rook, and pawn travel along the edgesor diagonalsof a square.The king, queery ano knight extend thei different powers through the world. board by moving in anY ot eight directions, forming three different kinds of octagon'

terms wi*rin us. Vet it is the leastpowerful piecein one tl"q lt.u,l*" ment on the world board, traveling only elgnt olr-ecalons both squareand triangular lines in any ot achon' yet tne tions] The king is virtually hidden from the cannot be captured around it' The king ;il" ;;;;G"Ives when it is surrounded and is theretore totary but loies restricted from movement on the board'

g havThe queenis the most powerful pieceon the board' .ovement'in uoy of the eight directions.of in* "Jiiti"J fte.que;n ot t1e mLifestation. She is the mother goddess'. tne mateworld, the genetrix in her natural "square" realm/ ls r(egtna riallv confi-zuredworld. In anoiher sense she the widely traveling moon C."ii, O""Jl of the Heavens, * i-uy" t"flects the light of the sun' the king'


Each of two bishops, or elephants, remains on its starting color throughout the game and has the power to slide aiong divine diagonals through the world. [:r one interpretation, the white squares represent the path of intellect (cal1ed. jnanayogain India) while the black (or red) squares represent a devotional path of service through the heart (bhaktiyoga).


The two knights, representing the awakening spiritual initiate acting in the world, move by leaps of intuition along the sides of the thirty-sixty-ninety right triangle, the one found within a hexagon and which Plato called the most beautiful of triangles. The knight moves by altemating between white and black squares,head and heart, intellect and devotion. At most it can choose from among eight moves, eight directions in the world forming the comers of an octagon. It moves through the foursquare worid by its own efforts, the only move the queery the World Mother, cannot make for it. Only a knight or pawn can initiate the game by making a first move. Until then the world waits in quiescence. H aa The two rooks (castlesor chariots) are the only pieces permitted to move strictly along straight lines on the side of a square in the four cardinai directions, representing our physical power to act in concert wiih the world's maleriaL structures.

a The eight pawns represent ordinary men and women attempting to cross the board of life through seven grades of initiation, mirroring the purification of the seven chakras, or centersof motivation. The Pawn can be male or female,representedby the Roman Mercury and Venus or the Gre;k Hermes and Aphrodite, who merge as the pawn Hermaph-




The chessboard as the earthsquare, with the twentY-eight stations of the moon as its Peruneter.

rodite. A pawn experiencesonly the simplest interactions with othei piecesand doesnot seethe divine forcesbehind it, being aware of only one step ahead (with an option of two steps o:nits first move) It is the only piece that can move onlv forward, never backward, asall other piecescan' \'Vhen with a chailenge,the pawn has the possibilityof pre'sented worldly or divine action, probing straiSht ahead to the opposiie color or along a short diagonal of the samecolor to colcapture or conquer its shadow opPonentln an aclracent the world's ordeals a umn. \Arhen piwn has triumphed over 'the board'seighth square,the iti and reaches seventhstage, opponent's divine back row, it achieves a higher state, or " oituve." Paradise is then regained and the Pawn transis forms into any piece the player wishes. Usually the Pa'vvn made a queen ind thereby becomes a co-creator with her, although great chessmatches have been won by choosing to trans{orm the Pawn into an initiate knight. A complete game of chess is an evolution through a series of geometric transformations of position and power' Every board situation is a snapshot frorn an ongoing_process representing the characteristics of the moment, like the hexagrams of th e I Ching.Each combination of pieces in various positions on the board exhibits the physical configuration of a particular energy Pattem. Use your imagination to visualize their relationships as lines of force composing an energy web. One way to play the game is to work with the geometric tensions and make moves that bring the situation into harmony. Lr the cosmology of chess,the four-sided board synbolizes the "world mountairl" so we can look at it as four concentric rings of squares. The game's battle is often for the board's central area, the "high ground," the area of the Hindu temole known as the "seat of Brahma," the central chamber housing ihe divinitY. The board is ringed by twenty-eight squares and is therefore traditionally associated with the twenty-eight "mansions," or nightly stations, of the moon's orbit around Earth, recognized in many cultures. The number twentyeight is interesting in that it resulis ftom the sum of numbers and was called a one through seven (28 =1+2+3+4+5+6+7) "oerfect number" since, like the number six, it is one of the



few numbers known to be equal to the sum of its divisors (28 = 1+2+4+7+14).Itwas also known as the number of days in a traditional lunar month of four seven-day weeks (28 = 4 x 7), thirteen of which make a lunar year of 364 (= 13 x 28) days, the time the moon takes to pass through the cycle of the zodiac. Thus the chessboardis circled by the moon in the same way that the moon's orbit enclosesthe squared circle containing the Earth (seechapter six). This is the limit of our action, the sublunar world, the "ring-pass-not" beyond which we cannot move our pieces. The same twenty-eight-part symbolism has occurred throughout the ancient world going back at least asfar as the Egyptian civilization. In terms of measure, the Egyptian cubit equals the width of seven "palms," or twenty-eight fingers. This symbolism extended to a colossal statue of Osiris, the Egyptian Lord of Cycles and of the Moon. Found near the Sphinx, the statue was made of twenty-eight pieces, the moon's monthly joumey. Later, in Britain, twenty-eight boroughs were arranged to surround the square mile of central London as part of the geomantic system mirroring the cosmological pattems on earth.

Full Moon

[,ost Ouortsr

First Ouortcl

NeuJMoon The phases of the moon



'l35 =.1080 xB The eight 135' comer angles of an octagonadd to 1,080, the radius of the moon in miles.

Octagonal drinking fountain

the game Since the moon is a symboi of reflected light' .l;;; slnnbollcatty tales place within the t*it "j :h: "f moon, as images reflecting arche$pal Pattems tl* Yt:?i wrth the geometryhasoften beenassociated and eight-side-d -I'hrouehou moon,"with water, and with lunar deities' t the world we find the octagonsymbolizing the the eisht iraior phases of the moon's cycles showingyin-yang or.tne of mirroi s).'rnmetry light and dark like t}:.e of trans{ormatjon' I Ching n eight trigrams, or stages Althoulh the cbnnectionbetweent}e moon and the number it" verv subtle, it indicates the depth to which the "inf',t *u,il"*atical philosophers probed into the archeuri"i"tt, typal pattems and found them in nature lf we look at a reg135 J1!, o.tugon we find that each comer angle measures Junru"r. ihe sum of all its corner anglesmust be 8 x 135 or degrees.This number is the radius of the moon in 1,0"80 miles, ttie sacred measure used to coordinate the cosmic dimensions and rhythms with the structure-function-order oi terrestrial society.The moon's associationwith the tides and water accountsfor the use of the octagonal shape m Christian baptistries, fountains, and fonts used in the ritual of anointing with water. John the Baptist appeared in ancient Babylon aJloanttes the Fish, who baptized initiates in the waters of their own unconscious, the lunar world "of reflectedimagesand archetypalpatterns,in preparahon tor ';ftte" with Shamash, the sun god, and direct baptism bv vis'ion of ihe spiritual light "behind" the archetypes' The game of chess, a much later development, -represents the to purify the iourney within our own psyche and the battle unconscious. to tarJust as gold is the metal whose color, and inability it with the sun, silver is nish, has tiaditionally associated linked with the moon. Indeed, on a clear night in the country the fu1l m6on appears quite silvery. The-ancients chose a sy'rnboi more apprbpriate than they may have known lt was only in this cenfury that scientists probing the atom found that the atomic weight of silver, its weight relative to carbon, is 107.870,or nearly 108, a tenth of the moon's radius of 1,080miles. These numbers keep recurring not we because make them do so but becausethey are inherent proportions of nature that expressthe timelessmathin the ematical archetypes.



LUNAROEOMETRY ART IN Idhile some cultures seea "man in the moory" more universal is the image of a "hare in the moon." It was the hareheaded Teutonicmoon goddessOestrawho gave her name to Easter, the celebration of resurrection and renewal that mirrors the moon being rebom by shedding its shadow. This eight-sided bronze mirror back from the Tung dlmasty depicts the "hare in the moon" using a mortar anJ pestle-s1'rnbols of male and female-to pipare the legendary "elixir of immortality." Its octagonal geometry implies the moon and renewal. Place tracing paper over the picture and connect the corners and crossings of the intetlaced squaresto seethe guidelines its designer used to achieve its intemal propo;dons and placements. As we ciimb toward the Decad we see how the principles of lower numbers serve as foundations.The principles of eight, the Octad, are a combinatjon woven froni the principles of its divisors, one, two, and four, the Monad, Dyad, Tetrad.The Octad displays wholeness,cycies,polarity, and manifestation. It serves as the principle of self-renewil at a

Representation of a lunar eclipse wiihin an octagon (Apianus,Astronomicum Caesareum,7540)

Tung dynasty mirror back

Aztec calendarstone



higher stage.The study of eight, therefore, gives insight into the mvsterv of the "same but different." the Oitad's role in the manifestation of the world is intertwined with the principles represented by all num-bers, not iust its divisors. With the introduction of the Octad into the universe that we're q.rnbolically constructing, we gain the ability to apply self-renewal and limitless growth' Wdre limited only b1rthe ever-expanding horizon ahead of us. But what is the nature of this horizon? Wherc must the Octad lead us?


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l|uilron Ihe Playing under the earttr, nine winters long, we grew mightily. -Eddic poem Nine worthies were they called.


And when Abram was ninety and nine years old, the Lord appeared to Abram, and said unto him, I am the Almighty God; walk before me, and be thou perfect. -Genesis 17:1 And about the ninth hour Jesuscried with a loud voice, saying, ELI, ELi, LAMA SABACHTHANI? that is to say,My God, my God, why has Tttou forsaken me? -Matthezn 27:46 Facing a wa11for nine years. Neither being nor nonbeing stand; the universe is ultimately empty. Nine years facing a wall, who is there that knows? -Taoist poeffi Possessionis nine points of the law. -Common saying

The Birth of the Nonagon. 501



THE WHOLE NINE YARDS We've traversed the mathematical landscape and have come to.the number nine, the greatest single-digit number within the Decad.The last among the sevenPythagoreannumbers (three through nine), nine is the limit to which the generaive principl"es of number reach. The ancient mathematical ohfldsophers called nine the "finishing post" and "that ivhich b'tings completion." It is the final step before ten' the Decad, a foim of tire Monad and beyond the realm of number itself. Comoosed of t}ree trinities (9 = 3 x 3), the number nine ,"p""ru.i, the principles of the sacred Triad taken to their lrtinost expression. Nine was considered thrice sacred and most holy, representing perfectiory balance, and order, the supreme superlative. ln ancient Asia, gifts given in groups ^nine were considered most respectful. In fact, the Chinese of words for " gift" and "nine" are identical. Nine is tle final number having a specific identity lt represents the highest attainment to be achieved in any endeavor. Nine is the unsurpassable limit' the utmost bound, the ultimate extension to which the archeS'pal principles of number can reach and manifest themselves in the wbrld. The ancient Greeks called nine "the horizon," as it lies at the edge of the shore before the boundless ocean of numbers thairepeat in endless cycles the principles of the first nine digits. Nothing lies beyond the principies of nine, which the Greeks called the Ennead' Folk sayings have carried the essential nature of nine orrer t tut y cettt tries. We say " a cat has nine lives," after which its luck runs out. Thlesof European witchcraft recount how a cat may becorne a witch at the age of nine. A "cat-ol nine-tails" provided s)Tnbolic maximum punishment to its victims. To "go the whole nine yards" is to go all the way in an endeavor. We say we're "on cloud nine" to exPressour highest joy. To be "dressedup to the nines" is to be dressed cornpletely and perfectly, omitting no detail. "A stitch in time savei nine" is a pithy way of saying that taking care of small problems as they arise enables us to avoid having to repeat the complete job (nine stitches) later. The ninth level is the proverbial nth degree.

of But in theway a markyou me, bargain, l'll caoilon theninth part of a hair. -William Shakespeare



The number ninety-nine is aiso used to express the horizon of the Ennead.Landlords sometirnesgive ninety-nineyear leasesand occasionallyoffer a symbolic 999-yearlease to tenants. A traditional ninety-nine-year pdson sentencels sometimes imposed to s1'rnbolicallyconvey an ultimate terrn. Having completed his earlier way of life with idols, Abraham was ninety-nine years old when the Lord spoke to him. Islam acknowledges the ninety-nine Beautiful Names of God, the limit to which we can know Deity, represented by the ninety-nine beads of an Islamic rosary. The final word ol prayer, " amen," from the Hebrew " so be it," transforms in ihe Greek numerical alphabet, the language in which the New Testament was written, into the number ninety-nine, signifying the highest extent to which human Prayer can ascendbefore the infinite mystery of Deity. Nine represents the bouldary between the mundane and the transcendental infinite. saviorsand heroes U/hen mythotogical gods,goddesses, went to extremestheir actions were characterizedby nineness.Nearly every tradiiion, from Northern Europe and Africa to shamanic Siberia, Asia, and the Americas, uses the nurnber nine to express an ultimate extent, journey, or duration. Northern sagasare repiete with ninefold s).'rnbolism. The Scandinaviangod Odin, ruler of the nine Norse worlds, hung nine days on the world axis, or Yggdrasil tree, to win of the secrets wisdom for mankind. Every nine years,fertility feasts lasting nine days were he1d.Nine Norse giantesses, who strode nine pacesat a time, lived at the edge of the sea and 1and.Prehistoric builders of the north recognized the symbolism of the number nine and built it into the strucfuresof sornestonecircles. According to Homer, the city of Troy was besiegedfor nine years.Odysseuswanderedfor nine more yearsafter the battles at Troy before he finally returned home. The Greek goddess of the {ruits of the earth, Demeter, who was often depicted wiih nine ears of wheat, searchednine days for her daughter Persephone beforelocatingher in the underworld. Persephoneis allowed to As a result of her experience., spend nine months of the year above ground when the earth is fruitful and for three months is compelled to live below when the earth is barren. The Greeksconsideredthe under-

Diagram of the Dogon society in Mali as a skeleton rnade of feminine cowrie shells-Dogon money. The eight clans (joints) intermarry so that the two numbers always add to nineihe number of the head, the chief.

more He rejoiceth of that than of theninety sheep, nnd nine whichwent not astraY. -Matthew 18:13



The nine worlds of the Odinic Mvsteries.

Engraving by Athanaseus Kirchener (sixteenth century).

world to be so deep that, it is said, an anvil dropped into it would fall for nine days beforereachingboftom. Tire ancient world's best-known mystery teachings, the Eleusinian Mysteries, were structured by the events of this myth: for nine days and nights the initiates took part in struitured ritual and drama duplicating Demeter'siearch. The Greeks also told that the birth of Apollo and Artemis by Leto took nine days and nights and finally came about only when Hera sent her a gold and amber necklace nine meters long. The virtuous virgin Artemis, role model for young Greek women, chosetwo nine-year-oldgirls as her companions. The Greeks honored rine muses,who personified and inspired the full range of the arts and sciences of humankind. As iJ to marvel at the results of their efforts, we accurnulatethem in museums. Tobe amused once meant to be under the sway of the muses. The Ennead was particu-


505 The Egyptian Ennead, or company of nine gods and goddesses,rePresentsatchetypal principles that regulate and rule the cosmos through the laws of number.

larly identified with the muse of dance, Terpsichore ( from the Greek terpsis signifying "twirleq" which iater became "delighter in dance"), who begins at the Monad and dances through the nine digits, spinning around at the limit of nine, and dancesback in this cYcle. Going back further in time, the Egyptians honored a company of nine "gods," ot neteru, their Ennead of nine manifesi principles, a trinity of trinities, which emerged from fouipairs of unmanifest gods to overseethe world's ongoing creative process. The Egyptians saw the whole of humanity, which they symbolized by the glyphs for "nine archers' bows," as subject to the ninefold rule of the Ennead. The pharaoh, their rePresentative on earth, was often syrnbolized by nine bows. This same s)'rnbol is referred to in the name of the vulture Mother Goddess Nekhebet, "she who binds the nine bows," or unites all people and so was called "the father of fathers, the mother of mothers." Ninefold rule is not uncommon in history. Lr rnedieval Euope, nine types of crowns in heraldry represented complete ninefoid rule. Ecclesiastical architects recognize nine styles of crosses. The Freemason for.rnders of the United Statesgovernment used the Egyptian Ennead as their model for the nine justices of the Suprerne Court, whose interpretation of the law is ultimate. The Egyptians honored ninefold companies of both deities and demons. The notion of ninefold cosmic levels spread through cultures they influenced and appears in others they never reached. In Christian symbolism wb find nine orders of angelic choirs in nine circles of heaven and nine orders of devils within nine rings of hell. Medieval tales refer to nine gates of hell (three of brass,.three of iron, and

came I hepharaoh forth of thethighs 'ftomb'etween thediaine Nine. -Egyptian myth



Nine-story Mayan temPle

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The3x3magicsquare appears in the cultures of Islam, Jainsof India, Tibetan Buddhism, Celts, African, Shamanic, and Jewish mysticism.

three of adamantine rock) with its nine rivers all pouring into the ninih ring, where Lucifer is confined' We aretold by Milton that it tooi nine days for Lucifel and his angels to fall tieir ultimate distanceftom divine from heaveryexpressing srace.The nin th represints the ultimate step' We learn that i't the ninth hour oi the clock Jesusdied and that afterward death appearsto his disciplesnine times' Nirie'was the central number of Celtic tradition, expressed oI as {hreefoldaspects t}retriple aytt 9f Y" Celtic maidens and nine virgins attendlng urldget' r ne sacred Beltane fire rites were atiended by a cycle of nine groups of nine men. The culture is filled with referencesto n1ne. Across the ocean the symbolism continues with the Native American, Aztec,and Mayan myth of nine cosmic levels (four above, earth in the center,and four below)' In this context,nine-story temPlesalso representnine levels of the underworld, the realm of our unconsclous' The ancient Chinese told a myth of the man who emerged from the Yellow River carrying a. geometric affangement Of forty-five points cal1edthe lo-shu,ot ivet plan. The groupings of points were recognizedas the numt"ts ott" through nine arranged in a 3 x 3 magic square' Its three columns, three rows, and two diagonals always add up to fifteen. Nine cells provide the smallest and first such airangementin which all nine numbers canbe woven in this form of matrix. The ancientChinesesaw the magic squareas a harmonious biend of the nine archetlpal principles of number and a paradigm of cosmic structure and process,mirroring the supremeorder of the universe.They used this arrangement to design the nine palacesof the Ming T'ang, or Hall of Light, in which the emperor lived and circulatedevery{orty days, just as terrestrial energy circulatesthrough the landscape in a year. ln the science of fAng-shui' the numbers within each ce1lof the magic square have specific significance for working with the earth's subtle creative energies for the good of society and the environment. The divine central acre was dedicated to Shang-ti, the supreme cosmic ruler. The magic-squaredesign was expanded and used as the layout of the capital city, with Old Peking at the center

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having eight avenues of access.On an even larger scale,it becanie the plan of the landscape of the whole of China, the Middle Kingdorn of the world. As the most auspicious number of celestial power in ancient Chinese,it orders nine great social laws, nine classes of officials, nine sacredrites, and nine-story pagodas. On the ninth hour of the ninth day of their ninth month was held the festival of the " double yang," honoring the ninefold creative celestial powers of heavery the archetypes represented by the nine digits. The 3 x 3 grid has been used elsewhere,notably in India, as the ground plan for temples dedicated to Vishnu, the Preserver. Each cell is subdivided into a smaller 3 x 3 grid, and each of the eighty-one (= 9 v 9; cells is dedicated to a different aspect of the deity. And we are told that a Buddhist monk's staff has nine rings. l:r chapter three we encountered the worldwide practice of repeating a phrase, prayer, or chant three times to enforce it with sacred intention. Thus, to lePeat any trinify thiee tirnes, fulfilling nine repetitions, represents an ultimate appeal. A famous example concems the three witches in Macbethrepeating their spell nine tirnes. A look into nearly every culture finds referencesto nine representing the ultimate extension. A superstition among many musical composers forbids the numbering of a symphony past the number nine. Even in modern times we find the principles of the Ennead expressed in secular affairs such as the American sport of basebali, played on the fourbasediamond, or square earth-syrnbol, fielded by nine players over a duration of nine irmings. The "bottom of the nin*u" the modern version of the ninth rung of hell, is a team's last chance to win, the ultimate limit. Beyond nine comes a new cycle. "Nine" and "new" are associated in various languages. The Egyptian glyph for "nine" was part of the glyphs for the suruise and new rnoon. The word "new" in many languages derives from the Sanlater the Latin noaa.The Roman skrit word for nine, naz.ta, market heid every nine days was called the zooendinae, and November was the ninth month of the Roman calendar (which became our eleventh rnonth when January and February were added to realign the calendar with the proces-

Nine-level pagoda

The weird sisterchandin hand, and land, Posters lhe sea of Thusdogo about,about, Thriceto thine,and thrice to mxnq And thriceagainto make up nine. -V\tilliam Shakespeare



and N owPeter lohn into wentup together the at temple thehour of prayu, beingthe ninth hour. -Acts 3:1

@oo@ Nine rings comprise the centriole of a human cell.

sion o{ the zodiac). A noaenais an act of worship over a a or period of nine days. From the Latin nonnes, Prayers, t the ninth hour (which worshippers begancounting at') emerged the word "noon," the moment when the hands of the clock and the sun are in their utmost, or highest, positions. There are few ninefold structures in nature. The forms that do configure the principles of the Ennead seem to be associatedwith the processof birth. The tail o{ the sperm half-cell is made of nine twisted threads. After it unites with the egg to form a complete ce1l,the first visible step in the doubling process of mitosis is the duplication of the centriole. Each centriole becomesa pole and moves to opposite sides of the cell while a spindle of chromosome tfueads forms between thern. Electron microscope photographs show the structure of eachcentrioleto be a circleof nine parallel tubes. The ancients could not, of course, have known about these discoveries of modern technology, but they attributed the nine months during which we develop in the womb and the nine openings of the body to the limiting principle of the Ennead.

T}IE FIORIZONNUMBER The source of the great esteemin which the ancients held the Ennead asthe principle of completion, achievement, and the end of a cycle can be found in the arithmetic properties of the number nine. The best example is the {amiliar multiplication table, which we force children to memorize without an understanding of its inner strucfure. On its surface the multiplication table appears to be a random grouping of digits. But it is actually a map of the cross-fertilization of mathematical principles. To seehow nine bounds it, we boil the table down to its digital roots. That is, we divide each term by nine and note only the remainder. Or we can simply add the digits together. The digital root of twelve is 1 + 2 = 3. lA/hen the first sum is a two- (or greater) digit number, keep adding until you arrive at one digit. Ior example,7 x 8 = 56 so fifty-six becomes5 + 6=11. Next we add 1 + 1 = 2. The digital root of fifty-six is two. This process/ called in




Nine boundsdecimalsof infinitely repeatingdigits: .11117...-+t/9 .. .22222. +2/9 .33333...-->3/9="t/3 .44444...+4/9 .5555O..._)Jly

The familiar multiplication table and the multiplication table reduced to its digital roots by the processof "casting out nines."

.66666...-+6/9=2/3 17n7...+7/9 .88888...+8/9 .99999 -+9/9 =L ...

2x9=18 3x9=27

8 1= 9 x 9 7 2= 9 x 8

medieval times "casting out nines," revealseachnumber's 4x9=36 63=9x7 essential nature and the principles of Monad through Ennead, which are at their core. 5x9 = 45 54=9x6 The table now revealsfascinatingpatterns.Eachdigit is highlighied as it recurs in the table to show its unique struc- Multiplyhg by nine teveals a ture within the whole. Some columns and rows include all mirror s)'nmetry among the In numbers. When dzy number nine digits but in different sequences. others, asin row si& only the same digits-three, six, nine-recur. ComPlemen- is multiplied by nine the tary rows, those that add up to nine-one and eight two and resulting digits always add to of seven, three and sit four and five-form patterns that are nine. Because this propedy the anclent Hebrews referred mirror images of each other but turned at right angles. to nine as the s1.rnbolof Only the pattem for the number nine has no reflection. immutable Truth, always proA square of four nines appears at the table's center,and then ducing itself and returning to a wall of solid nines forms a boundary along the table's itself, encompassing all numedges,the proverbial horizory or shepherd, which the numbers within its fold. bers below approach and revolve before in pattems but never passbeyond. Nine bounds and directs the choreography of the cosmic order revolving within it. Try expanding the multiplication table beyond nine and you'll see this same nine-cycle pattern of archetypes recur on larger scales. The ancient Hindus developed the mathematics of nine digits and zero to regulate the rhythms of their religious

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poetry, design temples, measure quantities and calculate istronomical positions to determine festival times. They used digital roots to check quickly the accuracy of large mathematical calculations, a process that was until recently taught in schools. For example, in the multiplication of 347 -J.4 and 1' + 4 times 86, the first term reduces to 3 + 4 + 7 = Ieavesa digital root of five. So does the secondterm, since 8 + 6 = 14 and 1 + 4 = 5. Their Product, 5 x 5 = 25, reducesto a digital root of seven. Thus, they knew that the answer has a digital root of seven. And so it does: 29842becornes2 + 9 +8+4+2=25and2+5=7. The Erureadbounds the regular polygons. If we look at the sum of all the corner angleswithtn any polygon we find nine as its digital root. Nine servesto bound or enclosenumbers and shapesdespite their apparent differences. As we look acrossthe oceanof endlessnumbers toward an unseeable infinite, we really need not look past nine to find the horizon of mathematics. 1 I 2 3 3 4 4 5 5 6 6 9 3 4 5 6 7 3 4 5 6 7 3 9 9 9 9 9 9 6 9 9 7 8 3 4 5 6 7 I 8 9 9 6 5 9 9 9 3 9 3 9 9

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The ancient mathematical philosophers were keenlv interestedin the geometricshapes associated with diffe.ent numbers. lnto how many different s).,"mmetric shapes can you arrange nine coins? You'll find many ways, but four stand out the square, cross, circle and triangle. Arranged as a 3 x 3 square,nine points form a triple trinity, the highest expression of the sacred Triad. Just as the number three and its hiangle are firstbom complete and whole among numbers and shapes, threetirnes threeasnine is the final child of Monad and Dvad, the ultimate completeness and wholeness. Along with one (ever-present unity) and four, nine coins configure th e third and hnal square number we'1l encounter within the Decad. Like all squares, nine is the product of one number (3 x 3) and sum of two consecutive "triangular" numbers (3 + 6 = 9). Nine, the ultimate number, happens to be unique as t|re only square number cornposedof two consecutive "cubes" (1 + 8 = 9). The association the squareand cube with the number of nine tells us that the principles of the Ennead are linked with those of the Tetrad, archetype of "earth," materialization, matter, configuration, form. Nine represents the ultimate expression of birth. We often encounter this association in the worldwide s).'rnbolism of triple goddesses, each with three aspectsor articulations.



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THE WORLD LABYRINT+I A second arrangement of nine points forms an X, or St' a Andrew's cross,-creabing different way of expressingthe Ennead's bounding prirrciples. From this nine-pointed core we derive the secrif for constructing the mythic labyrinth found ass).rnboland architecturein Egypt, Sumeria,Greece' Crete, Rome, Oceania, Cothic cathedrals, and elsewhere' Laby'rinths designed by sirnilarly arranged five and thirteen points have been discovered. A labyrinth is not a maze' A labyrinth has a single rout that rhythmically spirals to the center. A maze offers multiple pa*s and is'designed to confuse the traveler uriless its secretis known. Like the spider's web, a lab)'rinth represents the,r-veaye of the world.-Our motions are bounded but guided by its walls. SometimesbuiJt underground, a labyrinth s)rmbolthrough the world, from profanized the initiate's passage ity at its periPhery to redemption at the sacred center-In this traditional labyrinth the traveler reverses direction nine times through eight rings, cycling in rhythms of left and right, outward and inward. Reaching the mystic center at the eighth ring, the pilgrim completes the octaweand transformJto a higher,br deeper,realm, the core of Self. piece of paper to create Follow thesestePson a seParate this labyrinth. (1) Draw the cross of nine points. (2) Draw a vertical anchorizontal line through the center' Draw two vertical and horizontal lines from each of four s)''mmetric Points as shown. The resulting figure is a ctoss.Draw a curve from the top of the center line as shown to the top of the line to its right. (3) Draw a cuwe up and around to the point in the upper right comer. (4) From the point at the uPper left draw a curve uo and around to the first line encountered at the

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P oor intricated soul! Riddling, perplexed, labyrinthical soul! -John Donne (1572-1631, English poet)

left. (5-10) Follow this procedure of moving counterclock_ wise to eachpoint or line and drawing a cloikwise curve to the next point or line encountered at the right. The result is the traditionaI labyrinth design. Initiates ritually traversing labyrinths chanted tones as they followed the path. The loit clu'esto their music are built into the_labyrinth's structure. Each ring of this labyrinth cor_ responds to a note of the musical octave. The order in which the ftaveler ritually traverses the different rungs determines the sequence of notes to be chanted as the ,6ons,, of that Follow. the path with your finger to fini yourself ].|nTft: oeng re(l mrough the sequence E_D_C_F_B_A_G_cr. Jf vou play a musicalinstrument listen to this sequence heaithe to labyrinth's musical theme.



This traditional earthJabyri5rth rePresents the initiate's tourney through the "underworld" ol the unconscious through a full musical octave to our spirifual core.

THE BIRTfI OFTHE NONAGON Yet another configuration of nine equally spaced Points, those arranged around a circle, gives us the comers of a nonagon, the nine-sided regular Polygon. From the nonagon we can constructits "star" versions. We can approach the horizon, but we never actually reach it. Like trying to trisect arl angle, a Preciseconstruction of the nonagon is impossible to accomplish with onJy the geometer'sthree tools. But because of the number nine's intimate relationshiP to tfuee and six, any approachto a nonagon must be based on the geometry of the triangle and the hexagon. Follow these steps to rnanifest a very good approximation of the archetypal Ennead as nine points around a circle. (1) Consftuct a hexagon within a circle (seechapter six), and inscribe its hexagram star. Choose one triangle as the nonagon's"seed." (2-4) Frorn eachof its three comers,draw



Islamic tiling pattem integrates the geometries and s)..rnbolismof the numbers three, five, nine, and twelve into one space-filling cosmological pattem.

two straight lines across to the opposite side through the points that cross the other triangle, extending them to cross the circle. Figure (5) shows all nine points and lines simultaneously. Figure (5) shows just the nine nearly equally spaced points. (7) Connect the points in sequence to consfruit the nonagon. (8) Connect every other point to form the dilated nonagram star, and (9) connect every third point to construct three noncontinuous friangles. (10) Connecting every fourth point produces the contracted form of the nonagram star.

THENNEAGRAM PROCESS Another way to examine the principles of the ennead is through the arrangement of nine points around a circle. For this we look at the "ermeagram," the name given to a



remarkable diagram introduced to the West in this century I. bv Georees Girdiieff, who claimedto havelearnedit in an ironastery in Central Asia' It is remarkablein that it ancient is a simple way of looking at whole events as an outer sequence 'Th" with hidden inner connections' shows the organization of any whole "on"ugru-its essentialprinciples We've seenthree event in terms"of outstanding archetypes associated with wholeness and cvclic procJss-the Monad, Triad, and Heptad, represented by the'numbers one, three, and seven. They each contribute to any whole event. The enneagram weaves their principles into one diagram.We'll constructan enneagramasweappry based it to a practi"calexample, the functioning of a kitcheo's' on the model by J. G. Bennett,a student of Gurdjieff Begin by constructing a circle with nine equally spaced points. thii is our Monad, a nine-spokedwheel representIn ine the cycle of the whole process. this example,a kitchen priput"iatd te*et u *"J1, th"t-tcleansup and getsready to repeatthe procedure. Put the numeral nine at the top, and number the points arormd clockwise. Draw straight lines cofflecting Points three, six, and nine to form an equilateral triangle representing the principle of the Triad, the three-Part structure

Neutral. Circumstances, conditions,place. KITCHEN

Active. Something that brings about transformation or usesthat which is transformed. DINERS

Passive. Something that undergoes transformation. RAWFOOD



iriherent in every event-the passive, active, and neutral aspects.The passive element undergoes transformation. It is the raw food that will be prepared, cooked, and served. The active element brings about its transformation or uses that which is transformed. In tlds casethe active element consists of the diners,the peoplewho will eatthe preparedfood. The neutral element is the kitchen itself, which allows the polar opposites to interact in a constructive way, binding them into a complete whole at a new level. Thesethree aspectsform the comers of the triangle with the raw food introduced at point three and the diners entering the sceneat step six. Tht kitchen is at the apex at point nine. Thesethree elements form the structure of anv kitchen. If we can identify the three structural aspects of any event-the active, passive, and neutral-we can begin to understand the way it works. With these three guideposts we can fill in the intervening steps in the cycle. Each two consecutive steps extend the previous corner of the triangle. Steps one and two involve the kitchen. Steps four and five involve the raw food. And steps seven and eight involve the diners. Our experience in a kitchen enablesus to fill in the stagesto composethe outer sequence events. of But the outer events also have nonobvious inner connections to other parts of the circle, to other stages in the process.We find them by applying the principles of the Heptad, the sevenfold harmony within whole cycles like the

Eating the meal s Servingthe meal z

, Kitchengets ' ready for work 2 Kitchen at work

DINERS Cookingthe food

3 RAWFOOD Preparingthe food

; TllE tloRlzoN


musical scale.The way that in seven seeksto return to unlty is for.rnd simply by dividing seven jnto one (one-geventh)' The result eqiril lqzaslt+zssr1'42857. .. / a rePeatingdec" imal cvcle of the six digits 1-4-2-8-5-7. This cycle contains multipGs of three,which form the points of all diilts excepl the Tiiad. Sevenand ihree,Triad and Heptad, eachsupplies to the process what the other is missing. The Triad-provides overali structure and balance but not the details of the but is devoid process. The Heptad virgin guides the process The enneagrambinds of the principle of balaniedpolarity. them together,integrating uil t'itte di$ts. (The circle of the enneagram itself represents the tenth di$t, zero.) Drlw straight lines connecting the remaining points in this sequence (1-4-2-S-5-n, and draw a line from point sevento point one, renewing thq cycle. Thesesix lines are yet another way of expressingthe principle of six circlesaround a central seventh,'thesixtayiof the week around the Saband the perbath, the six tlpes of pieceson the chessboard lines of the I Cluirg. mutations of the six hexagram These inner lines can help us to "think ahead" at any For stagein the process. example,let's say that we're at stage gettlng ttre kltchen ready for work. To determine what onJ, we'll-actually do there, notice the line radiating ahead-to point four. Itle s us to look aheadto point four and consider what we'll need to prepare the food. So we gather cutting and cooking utensil;, pans, spices,and whatever is needed to prepare the raw food. The line between points one and


The fraction 1/7 = .L42857I42857.' rePresented as a continuous Path within a nine-pointed circl. Seven is the only number that divides into unity leaving a rePeating trail of differeni digits'

Eating the meal 6 Servingthe meal z

- Kitchen sets ] ready for-work 2 Kitchen at work

DINERS Cooking the food

3 RAWFOOD Preparingthe food



seventells us to considerwhat we'll need to servethe meal, so we also gather plates, eating utensils,napkins, drinking glasses,and so forth. Having thus prepared the kitchery we proceedin the outer sequence point two and get to work. to From point two we seelines radiating to points four and eight. Point eight has us consider the eating of the meal, the specifictype of nourishment and any dietary preferences of theseparticular guests.It directs what we'll serve and how we'lI serveit. Point four has us look at what we'll actually do to prepare the food, whether we bake, grill, fry, or steam it. At pbint three the raw food is introduced, and we now can move to points four and five, where the food is prepared and cooked. Next, the diners are introduced, and at points seven and eight we can serve and eat the meal. Thery we clean up and reach the ninth stage, our goai of completing the cycle in preparation for repeating it later. Frorn each poin! or stage, there are lines radiating to other points acrossthe circle. These are guides to help us A *on maybequite think out the processby directing us to look aheadand back alonein thedesert he and to other stages.You can apply the structure of the enneagram to any cyclic whole event in your life if you can detercantracetheenneagram in thesandand in it readthe mine its inner Triad and the intervening stagesthat make the process a cycle. The Triad and Heptad are complements, eternal lawsof theuniaerse each containing what the other is missing and compre. . . If two menwho haoe hended in the nine-stagewhole of the ermead. Another example is the cycle of the year. But we must been dffirent schools in think of the yearly cycle as the ancients did, as a circle, relinmeet,they zuill draw the quishing our image of the year as a straight line extending enneagram with its and toward an incomprehensible "eternity." The ancients of helptheywiII beableat many cultures represented the year as a circle, or spiral, whose dates recur with certain characteristics onceto establish whichof and resonate acrosstime. themknowsmore. If we put January 1, the begiruring of the Westem year, -P. D. Ouspensky(?-1947, at the top of the enneagram at point nine, the distance Russian mathematician, between each of the remaining points marks a period of writer, student of G. I. Iotty days, a traditional milestone in fime reckoninq and Gurdjieff) mythological symbolism. The complete cycle compreiends 360 days, the traditional duration of the year. Consider the remaining five days as the Egyptians, Babylonians, Indians, and Mayans did, as intercalary days "outside" the circle of the year.



The inner lines of the enneagram show us how different parts of the year have inner connections.For example,perhaps you initiate an action on February 10. Taking action is like planting a seed that develops slowly, eventually producing flower and fruit in their seasons.Every process needs time to germinate and transform in its unique sequence.Its outer order around the circle based on the Triad shows us the event's familiar sequence in sidereal time. But the inner lines basedon the Heptad show us how our actionsripple and weave acrossthe circle of the year in harmonic ttme.Lines{rom point one,February 10,tell us that our action will have ramifications on June 10 and October 9. This systemmay sound far-fetched,but it works as an agricultural calendar whose Triadic structure of seed (January 1), flower (May 1), and fruit (August 29) indicate proper times for sowing, iending, and harvesting. The Heptad's relations between thern. In inner lines show the necessary this structure the Egyptian calendarrecognizedthe year as three seasonsof four months each-Ahet, seasonof inundaof tion and sowing; Pert,season growth; and Shemu,season of harvest and imrndation. A look at the festival dates and religious holidays of many cultures reveals an emphasis on forty-day periods, grouped around each of the nine points of an enneagram. The Aztec and Mayan calendars,which contained eighteen Eachof months of twenty days, work nicely asenneagrams.



its nine points spans two months, so the inner cross-lippling of eventsand festivalsthrough the calendarbecomes easyto work with.

o Reaching the Ennead brings us to the peak of the first nine principles and archetypes. Each encounter with the number nine in math and myth shows it surrounding or enclosing and binding the essential elements of any event into a whole. It representsthe zenith of possibilities for number and the full duration of measure.T%efinal pythagorean number, nine, as the fuIfillment of the Eruread,completes the archetlpes that manifest the forms and processes nature. if With nine we exhaustthe world's pow;rs. As if after nine months in a womb, we are now ready for birth into life beyond which we have known. \Atrat remains for us is to so beyond the horizon, beyond the realm reachable numb-er by ano measlrre. The clue to the way beyond numbers is found in the fourth configuration of nine points arranged as a triangle. The missing point implied at their middie will be ow pith through to ten, the Decad, the culmhation of the coimic process/ not considered a number but existing as an empyrean realm beyond them.

o O O



o o o o


hn Itrnn

lltltnhe Beyrttd Leaving the old" both worlds at once they view, That stand upon the threshold of the new. -Edmund Waller The activity and the essenceof the number must be measured by the power contained in the notion of 10. For this (power) is great, all-embracing, ail-accomplishing, and is the fundament and guide of the divine and heavenly life as well as human life. -Philolaus I will sing a new song unto thee,O God: upon a psaltery and an instrument of ten strings will I sing praises unto thee. -Psalms 1.44:3 My skength is as the strength of .tery Because ' heart is oure, mv -At|rea, tord Tennyson (1809-L892, Englishpoet) I could not tell nor name the multitude, not even iJ I had ten tongues, ten moutl$, not if I had a voice unwearyilg and a heart of bronzewere in me -Homer 'And he wrote upon the tables the words of the covenant, the ten commandments. -Genesis 34:28 There are ten Sefirot. Ten and not nine; ten, and not eleven. If one acts ald attempts to understand this Wisdom he shall become wise. -Sefir Yetzirah(TheBookof Crution, third-cmtury Kabalisttext)

The Birth of the Decagon and Decagram Star.

Gone. Gone. Gone beyond. Gone beyond beyond. Hail the enlightening voyager! -Sanskrit mantram 323



NEWBEOINNINOS bers swirling wittrin ttre ringbf the horizon sharply defined bv the num6er nine' But where are we when we pass the h'orizon?Ten takes us beyond the realm of number itself' above the fray of ordinary numerical interactions and geometric relationships. It is a irew beginning, a ioumey into limitlessness. Ten represents a recapitulation of the whole' It hoids within itself the two parents of numbers (one and two) and their seven children ithree tluough nine)' Ten is a portrait of the whole family of archefyPes gathered together, simultaneously displaying each of their principles. Expressing the a properties of all numbers,ten represents slrrergy,a wnole parts, beyond the threshold.of greiter than the sum of its iumber itself. Like one and two, ten was not considered by the ancient mathematical philosophers to be a "number' " Instead, it is the epitome of number, comprehending all numbers' principles as a mathematical promontory' This all-encompassing archetype represented by ten has been known since-the Golden Age of Greece as the Decad' The ancient mathematical philosophers were great Punsters ("receptaand played on the words dekn("ten") and dachns cle"f to'emphasize the Decad's Property of containing the whole. They also referred to the Decad as "world" and "heaverL" since it comprehends and harmonizes all numbers below it. Ten rePresents all-inclusive perfection. To This numberwasof old understand the properties of ten is to know all. The Decad is heldhigh in honor, a paradigm of the creating Process, encapsulating all the arihetypal principles of Monad through Ennead' It holds as for suchis thenumberof if within ten fingers all their arithmetic proportions and geofingersby whichwe count, mehic patterns that manifest as the cosmos.With the Decad . . . or thenumerals we have all that is necessary for understanding the conto increase ten,andftom struction of the orderly universe' . To understand the Decad's principles we must look to its there arithmetic origins. ln chapter nine we saw how the essence Weagain begina new of any number can be found through its digitai root, the sum round. of its digits. The digital root of ten is one, that is, 10 = 1 + 0 = --Ovid (43B.c.-17A.D., 1. Thus, the Decad flows back to unity, becoming a new Monad. What the Monad is to the archetypal principles and Roman poet) numbers through nine, the Decad is to all numbers that fol-

oi the-oceT 1"*to the Aswetake steP tenwePassbeyond

o o o o o o o o o t,,


325 Two hands traditionillY sYmbolize the cosmosin its entirety, summed uP by ihe Decad of ten fingers. The left illustration shows the blessing by the hands from the mystical traditions of the JewishKabalah.Letters between the ioints symbolize relationshipsbetweenarchetypal number/letter pattems, the human body, and the cosmic structure. fihe right hands show Christian saints in each of twenty-eight finger a sections, lunar reference.

1ow it. Multiplying any number by ten is like multiplying ii by one.The ieiult is essentiallyunchanged,but the n9mb91 is brought to a higher level, an expanded version of itself' The Pythagoreans referred to ten as the "higher unity wherein the One is unfolded." Another way of approaching ten is by looking at the origin of the word. The English word "ten" itself derives from the lndo-Europeai dekm,signifying "two hands," a referenceto our ten fingers,the most versatileparts of our body and the only ones capable of reaching any other part. From *ris came the Sanskrit dasa,tll.e Creek deka,and the Latin The Germanic branch changed the "d" to a "tz" decem. sound (" zehn"\ and from this came "ten." The clue of "two hands" shows us the factors that interact to generateten. Since 10 = 1 x 2 x 5, it results frqm the interplay of the Monad, Dyad, and Pentad, that is, the Parentsof number (one and two) themselves fertilized by the principle of the regeneration(five). Thus, the Decadrepresents Power to infinite. Multigenerate numbers beyond itseif, toward the plying any number .by ten does not change its essential nature but only acts to expand its power. Fifty is an exPression of five. Ten is the Pythagorean "perfect number," slrnbolizing both fulfillment and new beginnings. The Decad, ten and its multiples, brings a1Inumbers to perfection and completeness. Thus, we praise something as "a perfect ten" and assign10 x 10 or 100percentas the highest grade on exams. Ten always represents the next generation, a quantum, or "logarithrnic," leap ahead. \Vhen we want to exPress absolute comparisons and contrasts we say something is "ten times better" or "ten times worse" than somethingelse and an army is "decimated" (literally "reduced to one-



tenth"). \nlhen w e reler lo a round number we often irnPly a multipie of ten. Likewise,we grouP yearsin multiples of ten ter. and nime them decades, of wh:chmake a century and ten of those which conipose orte ffiillennium' Celebrations like

of "jubilee"wereheldin multiples fifty t= | - 191 theBiblical (= x 10) years.The Egyptian sedfestival occurredevery30 3 years of a phiraoh's reign to renew his spiritual powers' . Modern mathematicswas greatly advancedwhen, in the Middle Ages, a greaterneed for calculating left the clumsy Roman numerali behind and became based on the decimal system of ten digits and place-value notation (of ones, tens, hr.rndreds, and Jo forth), which is easy to maniPulate' Yet decimal counting systems based on the ten fingers of our hands, the classic foundation of all counting rnethods, have always predominated around the world. Ancient cultures often had s)'rnbols representing large multiples of ten and methods of grouping them to express long durations of astronomical cycles and large quantities of weight, length, area, and volume. Even cultures without place-values (Babylonian, Egyptian, Roman), which used different s)rmbols io represent numbers, grouped them by tens.The tenth, hundredth, and thousandth numeral are each unique, always taking the system to a greater level beyond the previous limit of nine (99, 999, on) with a new encompassing s;zmbol.In modem form ten is the first value of two dlgits, and powers of ten (100, 1,000,and so on) extend another digit place, and these extend the archet1pe to new beginnings.

ln counting systems worldwide, each tenth step begins a new level and recapitulates the whole. Number systems reveal a.culture's picture of the cosmos.




T II TN wedgesare Babylonian groupedin powersof ten and srxty.

v 20 ((


ul TI






I2 {lY



<< ({ (tffi ((





327 Egyptian iine groupings led to the arch of tery also represented by the glyph of the penis, indicating the Decad's generativepower to create nurnbers beyond ten. Other glyphs (spiral, flowering plant, finger, frog [reference to m)'dad tadpolesl, man in praise, and the etemal circle ioined with a line) represented the greater fertilizing

{ | | tl l l l 2 3 4 6

itl | |




n 10

-D8, I 10

l h f o : : o o

100 1,000 10 ,000 100,000 1,000,00010,000,000

powersof ten'

O. 0 1

o o

l .l:l:l!il| 5 6







Mayan dot and line represent one and five. The oval "zero" brtlll powers of ten.

The famfiar Romannumeralsrecurin powersof ten. IIIIIIIVVViVIIVIIIIXX L=50/ C=100/ D=500, M=1000 NUMERALS CHINESE..JAPANESE | 5 + 1 g






+ 6

3 2 4A

1 < r o o s/\



328 The Hebrew, Greek, and Arabic alphabets served as numerals arranged in a tenfold pattern of exPansion. Each of these cultures encoded their great literature with the traditionai number symbolism of ten archetyPal pattems. This vertical arrangement o{ the Arabic alphabet also reveals the symbolic relationship between letters, numbers, and the four traditional elements.


Lettersas numerals





il I

't D ' i d

t I

t! !t

t (












- s 9

o t p 1 6 0 t $ l q

' 4 1 v ! o

? n

e A

l o o - g o o P a c o ? / y . ' f r Greek

el l

v../" / 300

'5 700

.? 6


.80 ' , ' /so

c 400 800

2 E 3 Earrh



e/ 7

|, I rOO 500



,'999,/ 1OOO

.J I 30

) 70 200

i 600




THE PAST TENS In ancient cultures nurnber s)'rnbolism was consciously and consistently used to express alignment with the cosmic order. Throughout the world, myth and religion are replete with examples of the Decad as a s1'rnbol of fulfillment' of expandedpower, and new beginning The appearance ten often represents the completion of a joumey and a retum to the origin after a purifying ninefold experience. The Baby_ Ionials celebrated spring festivals at the tirne of renewal and rebirth by honoring their deities in a festival lasting ten days. Demeter searched nine days for her daughter Persephone, finding her on the tenth; the city of Troy was besieged nine years and fell in the tenth; afterward, Odysseus wandered nine years and returned home on the tenth. After nine days clinging to the cosmic tree, Odin gained wisdom for mankind beginning on the tenth day. After Dante toured the nine rings of hell and nine spheres of paradise, his experience culminated with an ascent to the tenth, or empyrearL realm of transcendent spirifual beauty. In the Celtic traditions predating Christianity the burnirrg of the sacred oak yule (from the GaIIi'cgule, signifying "wheel") log took place on December25. This ritual represented the annual wheel of the sun's path across the sky poised at the winter solsticebetweenthe old and new years. The yule log was accompanied a stick of ash wrapped in by nine bands s)'rnbolizing the past year's nine forty-day periods. The nine bands were lit first, burning and bursting one at a time. After the ninth band burst, the ash-wood was revealed to cornpletely bum out the old year, which kindled the yule log soon to shine with the reborn light of the new year. The Bible was purposely packed with consistent tenJold symbolism to indicate fulfillrnent, perfection, a leap to a new beginning that comprehends the past. The tenth generation after Adam was Noal1 whose family became the world's new beginning of "one ianguage and of one speech." Ten generations later came Abraham, whose faith was tested ten times and who, at age 100 (= 10 x 10), accepted God's covenant, gave birth to Isaac, and began the monotheistic Jewish religiorg whose laws are encapsulated in the Ten

used I I heancients] " eaerflowing Nature"asa metaphor theDecad, for sinceit is,asit were,the and etterlasting eternal natureof all thingsand kindsof thing, and in with accordance it the things of the uniuerseare a and completed haoe and harmonious most

beautifullimit. -Iarnblichus

Andherbrotherandher mother said, thedamsel Let afewdays, abidewithus at theleast afterthat ten; shemaygo. -Genesis 24:55

Whm ongry, count ten beforeyou speak; ifaery angry, hundred. an -Thomas |efferson (17431826,third U.S.president)



untothe Giae thanks with the Lordzuith harp: psaltery theten-stringed unto doyesingpraises him' -Psalms 33;2

Commandments. The Bible describes how Eglpt was assailed by ten plagues before the Exodus began' Descriptions of thi deseit tibemacle during the {orty (= 4 x 10)years of wandering speak of ten-part structure and measure,as did Solomon's temple. The l1're with which David sang praises of God had ten strings to emphasize the transcendence of prayer. In order ?or Jews to transcend individual prayer and to worship as a group, a quorum, or minyan, of ten men is required to hold a unified service. Not only Jewishbut Christian, Buddhis! Sufi, and Islarnic traditiors have Decalogues:there are ten Christian spiritual graces (love, joy, peace,long-suffering gentlenesg goodness, flith, prudence, meekness,temperance)and ten Sufi veils, or obstacles,that keep us from seeing the world's and our own divinity (desire, separadorg hlpocrisy, narcissism, illusions, avarice, greed, irresponsibitity, laziness, negligence). The Pythagoreans recognized ten Pairs of fundamental cosmic tensions that puli tight the weave of the universe (limited/unlimited, odd/everu one/rrrany, right/left, male/ female, rest/motion, straight/crooked, light/darkness, good/bad, square/oblong).Buddhistsrecognizeten perfections. The ancient Egyptians, whose Ppamid Texts describe the Kingdom of Osiris (nature as perpetual cycles) upheld

The Endless Knot, one of the eight sacred emblems of Tibetan Buddhism, forms ten enclosures.



by ten pylons (the ten archetypes), described ten qualities of characteristic all created{orms (activity,passivity,powe\ position, peace). quality, quantity, relation, time, substance, They divided the sky into thirty-six ten-degree arcs around the earth and honored the archetypes with a ten-day week. Religions sometimes require a "tilhing," or relinquishing, of 10 percent of one's income for the group. The reader can find many more examples of tenfold s).'rnbolismin the religions and myths of virtually every culof ture. The consistency this type of symbolism is not due to a cross-cuituralsharing of information; it occurs to anyone who looks deeply enough into the properties of number.

are Th, Sorrrd Sefirot Tenasaretheir numbers. Theyarethe tenfingers of thehandsfiae , corresp ondingwithfioe. But in themiddletheyare lorotted Unity. in There ten Sefirot. are Tenand not nine;ten,and If not eleoen. oneactsand attempts understand to he this Wisdom shall become wise.Speculate, applyyour intelligence, and useyour imagination continuallywhen considering themsothat by suchsearching the Creatormay bereestablished uponhis throne. -Sefu Yetzirah

THE TREE OF LIFE One of the more well-known cosmologicalmodelsbasedon the Decad is the Kabalah, the ancient system of mystical Judaism. Students of the Kabalah begin their sfudy only after the age of forty (= 4 x 10), i.e., after the fulfi11ment of worldly concerns. It is an intricate system of letters, numbers,and soundsdescribingthe shrrctureof the cosmos/our body, and our inner life. The Kabalah approaches the twenty-two letters of the carefully constructed Hebrew alphabet and the mathematical archetypes from one to ten as divine instruments of the cosrnic creating process.To thoseinitiated in its wisdom it servesas instruction for consciousself-evolution. The complexity of the Kabalah is also intended to keep it out of the hands of those not purified to a degree that will allow its proper use.In fact, the word Kabalah is at the root of the modernword" cabal,"signifying a secretgloup. Although it was originally an oral teaching, some of the Kabalah's precepts were written in a third-century text called Sefir Yetzirah,or Bookof Creation,which describesthe unfoldment of the universe from the unknowable Ain Soph ("without 1imit"), the deity eternally beyond name, form, or description, the mysterious zero from which the creating processemerges.Always playing on words, students of the Kabalah point out that the word for "nothing" (aln) is composed of the same letters as the words "I Am" (ani). Thtts declaring its Being, the Unknowable utters ten



Crown' words, or archetyPes/that generatethe universe: ictory' wirao*, Underitanding, Mercy, Power, Beauty' V. ten "words' or Glory, Foundation, Kingdom' These Divine Sefirot,are said to be the;potencies" by which the ot tne manifestsvisible form. They are the ten archetyPes as ten lights or globes Process,deicribed "ot-i. "."uti"gipiettdot that descend from the Divine lrrrr,inort oi u.utrg"a as three vertical columns (10 = 3 + 4 + 3)' The namei of these columns were Wisdom, Beauty' -and itrength, after thoseof Solomon'stemple' and w9r9 laigr Arranged ln tnrs taken-asthe motto of the Freemasons' manner the ten Sefirot are referred to as the Treeof Life' or Tree of Perfection. The Kabalah teaches that we can only know the Unknowable God by his ten lights in the world, ten .varjcolored emanations of divine qualities, Iike the visible flames from an unseenbuming ioal' In our experienceof the world we cannot directly envisagethe source ot these liehts but onlv thejr effects,the con{iguratingmanifestaMonad through Decad'They are tiins of the ten archetypes, which the world's changesand transthe vehiclesthrough formations, the cJsmic creating Process,occur' Through deep study and contemplation of the ten numbers and an alohabet desimed to correspond wiih the Divine unfoldmlnt, studenti of the Kabalahgradually grow to know both the world and our deeper Self, finding them at one with their Divine source. According to the teachingsof the Kabalah,the first emergencefrom th-eUnknowable is the Monad, called the Crown' into a polarity, the Dyad, known as Wisit further sepurates dom and Understanding.Thesethreeform a divine triangle that gives birth to the seven archetypesor Sefirot, like the pareits of number generating the seven numbers three throush nine. Tils divine upward-pointing triangle atop the Tiee of by Life manifeststhe remaining archerypes reflecting itself triangles, un{olding the world of ten as three inverted spheres in four triangles, or levels, of increasing density, in other words, the foui ancientelements,or modem statesof matter, from electronic plasma to gas to liquid to solid (see chapter four).


The ancient Jewish seers who developed this model show its interrelation with the alphabet to produce thirtytwo connections, or paths, along this ulfoldment. Thus, a divine descent along thirty-two paths to unfold creation plus thirty-two ascending paths returning to Divinity produce sixty-four stages of transformatiory reminding us of the models of the I Ching and chessboard(seechapter eight). The Tiee of Life is sometimes overlaid onto the form of a human body, creating an image known as the Universal Man, or Adam Kadmon. The Tiee of Life reveals itself to be a caduceus, the modern medical s).'rnbol with roots in the ancientpast, depicting the three channelsof our subtle nervous system (seechapter seven).Thus, the Decad and Tree of Life areseenby the Kabalah'sstudentsasbeing within us, a world axis on which the cosmosand our consciousness tum. In India, this symbol is known as the spine, or shushumna, flanked by the rising and falling "serpents" of creative energy, ida and pingala. Along this Tree of Life are t},:.e sevenchakras distributed within the four levels. The system of the Kabalah requires a lifetime of serious study and self-development with a qualified teacher. It is one of many descriptions of the cosmic creating process based on the princlples of the ten archet)?es of number.



THE TTRAKTYS of Another cosmolo$icalmodel based on the completeness in four levels to describe the universal i"""o"ftnt for "ttftfdilg calledthe Tetraktys(from the Gre-ek wa! .;;"';;;;;tt ;?.".ro.ij"l i. ,lte Pythagoreanschoolaround 500e c lt consi"ts of t"o'poittts or pe6blesarranged as an.equilateraltritour anele like bowling pins, with one, two, three' and is Tetraktys pJnts in eachrowlsince10 =L+2+ 3 + 4' The 'a renient and powerful tool for examining simultane"on all ten archetlpal principles of number underlying the oJy geometry that maflifestsitself as nature' lt was held m tne fiiehest Jsteem, and, like the Treeof Life, it remained strictly in"the secretoral tradition for centuries before anything was written about it. The Tetraktys served as the basis of the Pythagorean school's studies of natural scienceand philosophy' Its triangular form.indicated td the symbolic geometersits divine denrncreasmg nature.Thev took its four levelsto rePresent and earth' the four fire, air, water, sitiesof the four elements, statesof matter (electronicplasma, gas,liquid' and modem ?ae a Coint solid). This descent from "fite" to "earth" representedby ,41 a a Line one, two, three, and four points is a metaphor for the way 244r"', a O a Surfoce themselves with mattaatt a O o Ovolume the ten archetyPes un{old and clothe ter to manifest as nature's visible Pattems' The Tetraktys also served as the core of their mathematical teaching. Its four levels show us the number of points in spaceof zer6-, one-, two-, and three-dimensions, thus defining all possibilities of point, line, surface, and volume' The Py-thagbreanssaw the cosmos unfold in the same sequence ai a pl-ant grows: seed, stern, 1eaf,and fruit.

o o o o o o o

o o

o the By studying the TetuaktYs ancientmathematicians noticed that terLis unique as the only triangular mrmber among the infinitely many that is a sum of consecutiveodd squarenumbers (1 + 9 = 10).

O O o o o o o o o o o o a o o o o o o 10


The Pythagoreans fust explored the gimp]g althrnetic and geometry in the Tetraktys. Usgg dark anf light stones (voulan use two different types of coins) to form Pattems among the ten points, they saw that the regroupings the re,real"ed relationshipsand patterns of interplay among numbers.But most imPortant, they observedtnat tne sirnple Tetrikws is a complete summation of three-dimensional forms becauseit hoids the geometry of all five Platonic volumes,which divide spaceequally in all directions (chapter form is ultimately basedon four). Everv tlrree-dimensional five volumes. Through a triangle of ten one or mori of these simple points the Decad's principles of fulfillrnent, completion, and wholeness are revealed to the symbolic geometer' It is also well known that the Pythagoreans were deeply interested in the rnathematics of music, the simple ratios among whole numbers that produce the pleasant tones of the the musical scale.The Tetraktys encapsulates numbers one, two, three, and four as the fractional lengths of a vibrating string that produce the natural seven-tonemusical scale: the octave (one-half and two-fourths) the double octave

o - f l - L

a a . a . a =

o o a O O O

a a a

a a a o

Q = . l 0

o a a o o o a a a a

o a a a o a o a a o

a o o a a a a c a a


a aFt

.@. Cube (earth)



The Decad exkudes into three dimensions as a tetrahedron when ten spheres,like oranges or cannonballs, are stackedinrows of 1 + 3 + 6.

Icosahedron (water)

.ffi. Dodecahedron (heaven)




Tetrahedron (fire)

.@. Octahedron (air)



(one-fourth), the musical fourth (three-fourths), and,the musical fifth (two-thirds). These produce the "spiral of fifths" or "spiral of fourths" and endless octaves of tones (seechapter seven). ' Members of the Pythagoreanschool were so impressed with the richness of this cosmological model that they swore oaths "by the discoverer of the Divine UnregenerateTetraktys, the jpring of all our wisdom, the etemal root of Nature's fount." The Pythagoreans realized that the Tetraktys represented an-ensehble, a unity, a summing up of the whole, that comprehended the completeness of mathematics and the archetvDes that manifested themselves as the visible ten forms of the world. A11 archetypal number principles can be studied in it. For this reason mathematical philosophers have studied the Tetraktys for over twenty-five centuries to investigate the ten mathematical archetypes that construct and guide the natural course of events'

THE BIG T.O,E. Mathematicians, scientists, and philosophers never stop looking for a unifying cosmology, a simple but complete scheme of nature, a theory that explains in one symbol or equation the universal creating Process. Yet it seems that whenever they do look deeply enough, they come to a bedrock of ten aspectsand rediscoverthe Decad regardless of their counting system, since it is a result of the interplay of the eternal nurnber itself. A recent wave sweeping theoretical physics has been called The Big T.O.E., or Theory of Everything. It ProPoses that the fundamental building "blocks" of the universe are neither "particles" not waves but infiniiesimally small "loops" of mathematical " sftu:.1" that roll, rotate, twist, vibrate, and tweak to ultimateiy result in the familiar configurations of matter. Only the shing's mode of vibration produces the universe's endless variety, determining all possible particles and forces of nature. The interesting poini is that the design of these "strings" requires a mathematics that only makes sensein ten dirnensions.



that each our 'of Simplvput, string theoryProposes (lenglh'width' ond s <it'space frn'iiiriiL;;' dirl1er,"siot ab to enlolded aPpear dimensions three is heieht) actually Triad' a a

as dimension is treated mathematically ;"=fi This Triad of Triads makes Rtdtripartite unity' il;;.;;"" of time accounts ,rirr"-ai-"nrio.,"" oi.pu"" in ail' The flow and archetypal f* ifr"-i""tn amensi'on. True to its ancient a s1,rnbolism, Decad describesa whole' beautiful $/hile string theory reveals breathtakingly theoretical evidence Jor there is no *"ti-t"t".ti"tf -9 to impossible.detect "1]:1s is presently r"-rJ,arty"q:"ationi ii. mathematonlv in the uncannily consistentequationsof itsthe first -i"""tti-gutioit i."'irtlnit" Pythagoris is credited with conducting to actual events by to" apply mathematics ing" numericat iatlbs among musical strings' in string find theorii we are witnessing fifth-century Bc mathematlcs findi-r{grebirth at the end-of the twentieth century' along with J recurring recognition of the archetlpal principles found at the core of number and shape'

n&u& String theory Postulates a . ten-dimensional universe rn which each of the three dimensions of sPace(length, width, and height) are a tdnlty, plus the dimension of tune.

Each thingis oftike form THE BIRTH OFTHE DECAGON The geometry of the Decad emergesfrorn that-of the Pentad -shares its arithrnetic Pattems and self-regenerating and golden ratio. The fact that 10 = 1 x 2 x 5 iells us that the decagoncanbe constructedasthe doubling of a a cirile. I lhile the ten-sided decagon is born through the expressthe piscis, theDecadcomprehends geometries uesica including that ing all the archetypes from Monad to Ennead, of its own "parents." FoIIow these stePs to construct the decagon and its internal patterns. (i) First construct a pentagon and pentagram star (see chapter five). (2-3) From each of the five corners draw a stralght line across the circle and through the point wLqre lines intersect, extending eachline to reach the circle. (4) The five comers of the pentagram plus five points indicated by the extended lines produce ten equally spaced Points around the circle.

and et.)erlasting comes roundagainin its cYcle. -Marcus Aurelius



pointsarourd thecirclein diltr ,Cormecting -startins ".-r7r-lr-_yrelds. drnerentconfigurations the Decad,(5) of wrth the topmostpoint, connect consecutive poinh t; structthe decagon. Conn, (6) ";;o


P-^"i::]" two

two discontinuS;; i;,;;":li::il the"decagonat i",:";:"pers

dj-qcontinuous pentagram stars. you can use

Jtar.,and (st";;.y f.,1rth;fii;

::i:;fft :: tr,::


toexplore itsielr-reproduc;s o;';;

Use your imagination to create and explore . its other rntemal geomefic pattems by drawing shaight lines oerweentheseten points and points created at inte;ecfions. r nesepattems arethe onesusedby artists, craftspeople, and architects of all eras in designing *.rk;th;;5"pi; ,h" principles of the Decad.

@@ffiffi ffi U @W A



DISCOYERTHE DECAD IN NATURALFORMS The principles of the tenfold Decad manifest themselves in living nature, often in organisms that display fivefold_symmetries like microscopic diatoms and radiolaria. More highty developed creatures like lobsters and crafish, cousins of the eight-legged spider, have ten walking legs. The squid, the animal with the most complex eye in the invertebrate world (sornetimes mistaken for the eightarmed octopus), has ten tentacles. Sincethe most widespread expression of fivefold principles is found in the plant world, we can expect to find tenfold geometry there, too. While some flowers, like the wellare caerulea), considered to known passionfl ower (Passiflora five Petals subtended by have ten petals, they actually have five leafy bracts, giving the appearanceof ten petals. But tenfold arithmetic and geometry can be seen in their fruit and seedpods,as in many rosesand citrus fruits. Along with the Pentad's principle of regeneration, the Decad and tenfold symmetry are found where the plant holds its new beginnings and leaps to a greater level of fulfillment and completion of the whole,



TTIEDECAOONIN ART DISCOVER Construct on Istomic Titing Pottern We've seenelsewhere how the Islarnic tradition uses Seomeachof the universalarchetypalpattems,and etry to express this use extends to the Decad. The following construction guides you to produce a decagonwith an intrigui n g,intemal tiling pattern symbolizing the regeneration and,fulfiliment of enlightened humanity when it apPears on walls, carPets, books, and elsewhere. Ai1 lines should be drawn lightly until the outline is completed and the significant lines can be darkened to emphasize the pattern'

o (1) Construct ten Points around a circle. Connect every fourth point to produce two pentagramstars' (2)Draw short lines within and parallel to the central decagram star to "thickenf it. (3) Place your comPass point on one of the crossings (poini a) of the pentagram stars and open it to the nearestcomer of the inner decagram(point b). Lightly turn a circle, crossing the line at point c. (4) Now place the point of your compass at the original circle's center and oPen the pencil to point c. Lightly tum another circle, crossing the lines at point d and others like it around this circle' (5) With the straightedge draw a iine connecting points c and d, extending it siightly beyond them. This will create a small regular pentagon pointing toward the inner decagon.Do the same with every pair of points around the circle. (6) Notice where the ten extended lines cross. Join thern to create a large decagory the outer edge of the tile. This construction serves as the basis of the many variations of the tiling pattern and geometric ornamentation. If you wish to develop the tiie in more detail, notice that only two pattems complete the Islamic design: a small four-sided "kite" containing a smaller replica of itself and a Pentagram containing a five-pointed star surrounded by repeating irregular "T" shapes.With care it is possible to construct this pattem in the appropriate areaswithin the tile. Reproduce multiple versions of this tile to seehow they "fit" together by leaving " gaps" in interesting geometric cr'hmptrioc



TRESRICHESHURES Another of the many examples of the transcendent s).'rnbolism of the Decad is found in the illustration of Paradise in the famous ?}s RichesHeuresdu Duc deBerry in the fifteenth centuqr. The scene depicts impotant stages in the story ol Adam and Eve, from the serpent and the apple to expulsion frorn the garden. The composition is obviously designed within a circle, the traditional symbol of heaveq wholeness, and unity. The artist has actually left visible some of the underlying geometric frame with which the scenewas composed. When a decagor; symbol of the Decad's transcen-

342 e. Paradis Geometry alter Charles Bouleau (The Geometry) Painter'sSecret


dence and fulfillment, is inscribed within the circle, the internal guidelines become aPparent.Draw straight lines connectingevery other point to construct two mterleavmg pentaqons. Straight lines dranm in different Patterns reveal the scene'sinvisible grid For tetwe"enthe ten p-oints lines connecting the pentagons' comers example, vertical and ciossings define the horizon and the architecture of the fountain and also integrate the figures within the scene' Draw lines, or simply hold the shaightedge between comers and crossings to discover the reasoning behind the piacement of each element within the circle. Finally, Adam and Eve find thernselves expelled into anew beginning beyond the gate of the ten-pointed circle of Paradis6. Yet even their Placement in the world of toil emerges from the geometry of the decagon. The circle is actually inscribed within a square, implying that Paradise is imbedded and hidden within material manifestation. As



always,extensionlies along diagonals.Put your compasson the bottom left comer of the square and open the pencil point to its upper left corner.Tum a circle, or part of one, to see that even in their expulsion Adam and Eve are still embracedby the Paradisethey left.

THE CAT}IEDRALOF ST.JO+INTHE DIVINE We tend to think of cathedralbuilding as an art of long ago, limited to a flurry of cathedralsbuilt during a few centuries in the Middle Ages.But therestil1existsa small coreof peridedicated to preserving the European patetic stonemasons In cathedrals. fact,the world's largestcathedralis now in the processof being built entirely of stoneon the highest hill on Manhattan Island in New York City. The cornerstone of the Cathedral of St. John the Divine was laid on St. John'sDay, December2Z 1892,and the structure is expectedto be completed near the end of the twenty-first century. \iVhile built primary of iimestone,the cathedralalsoincorporateseachof the twelve types of stonementioned in the Revelationof St. john the Divine, the last chapter of the New Testament. Keeping the cathedral-buildingtraditions afve, masons and sculptors come to New York Ciry from around the world to pass on their ancient knowledge to young people who live in the community. But the stoneworkers' ancient techniques are now augmented by computers and laserguided saws,although the detailed carving and placement are still done by hand. From its inception to 1945,the cathedral was overseen by five architects, perhaps the most significant being Ralph Adams Cram of Bostory who was influenced by the French Gothic cathedrai haditiorL well known for the wonderful geometry of its architecture. While the architects left detailed blueprints for the cathedral's construction and the dimensionsof every stone,they,like their predecessors, left no clue to any underlying geometric schemethey may have used to integrate and harmonize the whole design. Like the Egyptian and Mayan pyramids, a cathedral is built from the ground up but is meant to be viewed as descending from heaven above, hanging down from the sky.

Proposedcompletion of the front (west facade) of the Cathedralof St. ]ohn the Divine.



Cathedrals in general-areamong the greatestsymbols of Divine manifestation in the world and of spiritual regeneration and retum to the Divine within ourselves' The length of this cathedral is 601 feet, the length of an American football field plus one footbail; more important, the number 501 is associited with the Greek word knsmos," cosmos," or "orderly universe." The geometric symbols of regenerative life and spiritual fulfillment are the pentagon and decagon, and expressionsof the archetypal principles oJ the Pentad. Decad. lt is logical to look at the cathedral's underlying geometry as somehow related to these shapes' The {ollowLg constuction will help you to discover for yourself many of"the geometric relationihips integrating the elements of the froit entrance, that is, the west facade of the Cathedral of St. john the Divine. The sameapproachwili allow you to although of investigatethe architecture Europeancathedrals, the seed-polygons will differ. 1. A rectangle has been constructed around the maximum width of the cathedral at its base and the full height to the tops of the towers. The crossing of this rectangle's diagonals show us the facade's center. A circle has been circtrmscribed around the rectangle. 2. Around this circle have been found the ten equally spaced points of a decagon, symbol of compietion and transcendence.Every other point is connected to form two interlaced and opposite pentagons. Notice how the base of one pentagon matches the cathedral's base, while the other pentagon limits the height of the towers, whose innermbst turrets touch the points made where the pentagons cross. 3. Slowly we can un{o1d the geometry of the cathedral. An upright pentagram has been drawn in one of the pentagons. Its crossings reveal the distance between the towers. The crucial placement of the rose window is {ound within a pentagram star inverted and lowered from the center of the large pentagram star.



Geometric shrdy a-fter Keith CritcNow.

The placement of other important elements and details of the facade are left for the reader to find by completing unseen lines guided by the points of the overall decagon. Becausethe plan includes the geometry of the pentagon, it may be " folded" into one half of a twelve-faced dodecahedron, the fifth Piatonic volume and another symbol of heavenly order. lts floor plan also does the same; together thev comprise a full dodecahedroru the heavenly sphere



rise In an instant, from Set timeand space. the nnd become a world aside . world within y ourself . -Shabestari (c. 1250-1320, PersianSufi poet)


completed. In effect, the entire cosmic creating process is summed up in the cathedral. Designing a cathedral using symbolic geometry comes from a timeless tradition of temple building and geometric design. The temple rnediates between humanity and the heavenof archetypeideals,displaying the geometryof both. The discovery of an underlying geometric scheme in architecture, alt, or nature can give us a glirnpse into the world of archetlpes. Within the imperfect and flawed actuality lies an inherent ideal invisibly guiding its way. The joumey from the parents of numbers one and two and, through the world of their children, the seven numbers three through nine takes us aiong a path of primal archetypes and their expression in the manifest world of form. But the step-toten is not so much a leap acrossa chasm asit is a recognition of the inherent unity and wholeness that have been present all along. The Decad seemsto observe from beyond the fray of numbers, enclosing them and enabling them to extend beyond themselves to higher expression and fulfillment. The Decad is the recurrence of the Monad, unity at another level. With this new begiruring we areback where we began,although the better for having gone tfuough the experience of which the Decad is the summit.


lhat l{uw [unstrur Yuu've . . . what have you actually done? If you followed th9 s]eps for each of the tonstructions,, you used the geometer's three tools (and your awareness)to represent the archetypal principlesof the Monad through Decad,one through ten, the circli through decagonlpluJthe twelve-sided dodecagonand the five Platonic volumes), and you've acquired some practical skills. These are the constructions used by architects, carpenters, graphic designers, surveyors, and engineers' Remember to begin each construction with a central Point while centering yourself within. Watch as each polygon emergesfrom the polar tensioncreatedby the two circlesof a ztesiiapiscis.WIth practice you should be able to reproduce eachconstruction from memory. You'Il seethat certain numbers and their shapes are closely related, such as two, four, and eight; and three, six, nine, and twelve; and five and ten. Their constructions grow from related ones, and their principles are so closely allied that they teach us about each other. The georneter contemplates these constructions and gradually becomesfamiliar with the principles they express by developing a feel for each geometric motion, the particuIar way it expands the design, and the unique relationships it producesamong the lines, angles,areas,and volumes. In contemplating each constructiirn the geometer becomes familiar with the messageof each mathematical archefire, getting to know it as a different expression of harmony. With


aim, The geometer's is therefore, to imitatethe olically, unioerse symb its depicting central paradoxby bringing of togethershnpes dffirent orders, uniting geometric themassimplyand and as accurately possible a thuscreating cosmic xmage. -John Michell



enoughexPerience geofi the

AI l\ aturewill rez.teal itself if we will only look. -Thomas Alva Edison (1847-1941, American rnventor)

AU noturr',diff'rence Keeps nature's aII peace. -Alexander Pope

eye within any storm.Nature manifests herself in her organic and inorganc forms the principles of numbers as if thiugh shaoi.,* frf t*, or templates.The few charactersof naturei ulptiubJt ,e"r. rn a consistent geometric design{anguage. All nature,s shapes represent invisible forces, fundamintal principles, and cosmic processes madevisible. Thegeometer, ;#;;step motions are a metaphor of the vJorld,s drarnu] Ti" orclerly process of geometric construction shows us the steps nature metaphorically uses to cloflre the archetv.oal patterns.The geometergradually leams to seethe worid'of corunon expertence govemed by laws of nature that can as De underctood as expressionsof mathematicalarchetypes. Look at the torms you are most familiar with, like planis, a llower, pineapple,or pinecone,and practiceseeingth"- ,ro, as "things" that simply popped into view but astlie result of a gradual process of manifestation. Imagine each havine begun as "invisible" light that shapesitsJlf irrto seomet.i? pattems of energy or lines of forcethat configure ire zuide_ lines upon which the visible matter pru"ilpitut"r. if yo., applied the geometric constructionstithe illustration; of nafural forms from leaf to flower and butterfly to starfish, you should soon be able to look at other natural forms and recognizetheir generalgeometricpattems. What characters m the geometricalphabetdoes that vegetabiein the grocery store,display? triangle,square,penta"gon, A hexagon]rpi.ui, or ouler shape/ lt your pencil was sharp and you further oeveropedthe constructjonsupon the illustrations,you mav have been awed at the degreeof detail to which njtu re foi lows her patterns.By leaming to recognizenature,s calJig_ raphy and observing the world throug:hits shaping princi_ Ples, the geometer can read nature like a booi and understand something of nature,s intention. Deep within yourseft you already know the cast of charactersand the pot, the principles of number and shape that nafure uses. The geometry servesto remind us of it.'you probablfk";; other examples of geometric nature not chosenhere, and now you have the skills io explore them on your own. A

thisskiiltorec ognrze character the orher.siruarions.rh;;";'#,:,:T:.:T:i j!",:"1,,.,,:';':'.:L;: "f-,"h:tl:::11-transfer



knowledge of geometry can enhanceour wonder at the familiar. lking natureaswe ordinarilydo is likew.a Aooreciating

But and beauti--fuI servisa localpurpose, jf we ny T u j:]

tnrou'glr a city ainight

amidst its various Ughts' hach [gnt.]s

abovethe city we can look donm and seethe pattelnsthat a.ll the lights make strung along roads and streetsto homes and commercial centers, comprehensible pattems we cannot perceive from the ground. We gain a larger view of nature, 'of tif", arrd of ourJelf, and we can appreciate the unity of variety. The ancient mathematical philosophers called. this mundi, the integration of all nature's Pattems tlte harmonia harmony of the world. '1Why are In a sense, this book has asked the question things shaped the ways they are?" We have seenthat: I Nature's designs are best suited to their particular purpose. By recognizingand understanding them we can interPret nature's problem-solving strategies in that situation. I Nature's forms are the most practical and functional and most efficient in terms of space, materials, energy, and time. Nature's patterns teach us how to get the most from the least' I All nature's forms undergo continual transformation. Every process strives for balance, harmony, and wholeness. Geometry reveals the process through which harmony cornes about. I All the natural events and Pattems that manifest themselves, whether we urderstand them or not, had prior approval o{ natural laws. Nothing mani{ests or occurs without the urriverse's consent. Seeing how geometry shaPes natffe, you can undel stand why ancient artists, architects, and craftspeople of many cultures were impressed by its power and its ability to ennoble human creations, The ancients were aware of nature's geometric language and purposefully employed it in their arts, crafts, architecture, philosophy, myth, natural science,religiory and structures of society frorn prehistoric



ancients for ciedit AI th.t,^l.iris;;iti,,lio:i: gllethe Ll o manfinds dfficutt it enrecurtuies ljl|j*T$)8;, Il I:y jlg:,*r *o ti in,1*^, tro rft to ttntare /,et except the man ruhohasdeserted nature, -Seneca the younqer (4 s.c.-55a.o., RomL statesman and philosopher)

lT::-lh*gt, Scnolars and

researchers ,

the Renaissance. world The



I he unionof the mathematician the with poet, feroor with measure, pas sionwith cone ss, ctne :::this surely is the ideal. -J{illiam James( 1842_191,0 , American psychologistand philosopherf

ITffi ;*'nq:ff ;Jlf?.{,ili;;T"#.t :{ f.: diversiry and in which we pariicipate, *"lti-irn,

to treat the soil, water,air,plants,and creatures rrrr"".-ii,,'-^*^l'^11-'jl','". rn ways bom of understandil:"^:1{f:t*lty1 "ooperatively, the whole' respectfoi its parts, compassio.,, ^1rpurpose' ur,4 Comprehending .,"tu.e'" tp'"J;ilil; "o,otl ;'nmon

ffi i"l.,in'?"1i":""'H" ;11,.",i*""q.i"'"#*:ilff ?,ffi fl1l:1 ;il;il.il1'ff#':;"H"T'f ffi l""d ff oecomegenerationsof adul j.:,T"Tl;iT; t,"irhltn"*t,ia.a"ii!',n'#Tiil,:T;;f re.lationship with our enviionme,ntal matuix narmony in all its forms, ir

BETTERLTVING THROUGH GEOMTRY and of roles rra_ ll.lT-t^,if .gitlpid change rranstion the of oruonal instifutions, *" lia,

and ;;; fi"Ti::*:ff"rselves o"' '"rutio"'r'ii'


We arestar-stuff conternplating stars. the -Carl Sagan

we arl nuq.",: p".;+;;;;,r,r"";Jl$il'j:'",tonas rind ro tearn to resee the world in terms of its pattems

.iit::! #,r j,T"lfi;lii::"" il;;;d;';? #,;:dtr 1i ':": i..:,. *;' to nottring lessthan our ow1 i-""r^,-"-,^I'l^ l:

,"uii,'g it, ;";';;;",TI,IF

resnapng our vision and co ahgned with tife-facts.,"r"1".3:!l,s

g:::!,,,'i,'tlrri*u'r",-uJT:::i1T"::T#f trJ'.7 a,new perspecfive

selves.It's rime \de, as a global whole, relinquish olJ_.J_ els of looking and ieamin"gand Uegmto cood"r;i". i,,"r*" rn nature's script dispels lhe

ii:L::*:.l:ffff!tiffi:il#;'-xlll*::l il [:*,

o";"i# "u"au"o ;;#"r#:?:ff;ll,";[;rfl?t;:;

Weoftenactasif innerh naturenuas unconnected ,"i*, o.rtu. .,utr;, ;;# i-T1n

larguage and



resuires a shift within us. But once this shift occurs and we seJ the familiar world in terms of its shaPesand principles' into sharper a ljeht tums on and the world brightens' comes through it" reli?f. everything speaks its purpose fltTT.ttgPi T Even without knlw-ing it we use the samedesigns a catneoral nature. Look at a microscopic diatom and see that ^:l:ut".t r"r" witta.*. Ultimately, the same energy us AU uruand suides the natural world does the sametor -" found in human body proportions' which i"L""iJ"t-i# proPofilons can we have se-en be repackagedto produce the and Salaxy Ltrs aslr of a crvstal,plant, animal, solar system' a smgle th" |-rtiu".t" is one single organism, motivated -by oower, developing in many ways to gradually become ihriugh the i*utettu"s of the creaturesand 5il*'oiitt"rr forcesit Produces. Above the entrance to Plato's Academy in ancient Adrens were the words "Let none ignorant of geometry enter here." A knowledge of mathematics was prerequslte to the school'Jteaching' But let us end this book io, "r,tty with a switch and say, "Let none ignorant oI geometry exit here."

Inaudible to our deaf mortalearc Thewide world-rhYthms woaetheir stuPendous chant Towhich life striaestot'it

here ourrwme-beats , our Melting limitsin the illimitable' Tuningthefinite to infinitY. -Sri Aurobindo Ghose

Credits Page


Caloin and HobbeJ, copyright 191 Watte$on RePrinted with permission of Unive$al PressS).ndicate. Divinity cirtums.ribing the univers, photograph courtesy of Osteffeichis.h Nationalbibliothek Bikr-Athiv PorFat'SammlunS Dagrood, rprinted with special prmission of Kint Features Syndicate. TheCrcationoJthe Wotld and lhe Expulsion Iron' Pandise, @uftesy of The Metrcpolitan M6elm 1975. {1975.1.31) Milk splash photograph, copyright The Hamld E. Edarton 1992Trust.


xxn 5 11 27 34 47 58 59 76 n 100 110 125-126 133 136 737 138 154 161 ' 164 175 186 r8S-190 ml 203 204 206 m6 m7 274 277 225 272 233 247 265 26 276 29 338 342 343,345

of Art, Robe.t I2hman colletioo

Ttu Churning of ttu Mi w Ocea fromT\e Lockwook Kipling Collection IM22-1917 by courtesy of the Board of Trustes of the Victoria & Albrt Muselm. Egyptian pectoral photograph courtesy of Albe.t Shoucaii Bulet cracking e8& photoSraph courtesy of SciencePhoto Library: Dr Gary S. Settlesand Stephen S. Mclnty. Pharaoh Amenhotpe, photograph courtesy ofjiirgn Liepe. Pectoral of Senwosret II, photoglaphcou(sy GiJr,1915.(16.1.3) of The Mtlopoltan Musm of Art, Purchase, Rogrs Fund and Hnry walters

Egr?tian vulture pmdant, photograph courtesy of The Eg,?tian Museum, Cairo. photograph courtsy of The Brihsh Musum (negative no. PS248968). bdy ol the Beasls, StiI hom H8, Noot, counesy of Republic Pictures Corporation. TheFamilVCkcus, copytigtt O Bil Keane Inc. Standing man, face, and cubit drawings, copyright O Libby Reid. Sun god tableL Babylonian stela, ourtsy of The Bdtish Museum (n%ative no. 8.2057). Khg Tut-Ankh-Amon Sold mask, photoSraph courtesy ofThe ESyptian Museum, Cairo. Egptian pectoral photographs, courtsy of Albert Shoucan Dispute ooet the Holy Sactanent,photograph courtesy of Monummti Musei Ca erie Pontificie, Vatican. Heart muscl drawing, .ourtesy of S.ientific Amrican, Inc. Huricane satelite photogaph, National Oceanic and Atmospheric Administration, U.S. Dept. of Commerce. me Bri.le af Ftaflkensteifl,photograph copyright O by Universal Cjty Studios, Inc. Counsy of MCA Publishing Rights, a division of MCA Inc. The Wizardofoz, photograph copyright @1939Tumer Entetainrnent Co. All rights reservd. Tm lover's knot geomeiry, counesy oljohn Michell. Line dmwings, copyright O Libby Reid. Division of ancient Ireland. courtesy of John_ Midell. A clerk, astronomer, and recorder, photo8raph couriesy of Bibliothque Nationak; Paris. Tepeephotograph, courtesy of lohn Sheppard. Daoid TryinSon Saul'sArftrrl photo8raph courtesy ofThe Metiopolitan Museum of Art, cilt of ,. Pierpont Morgm, 1917.(12190.399) Agus beforc the Oracle of Delphi, photograph courtesy of Antilensammlun& Staatliche Musen zu Berlin. B[a Coola sun mask courtesy ofDparhnst of LibEry Senies, American Museum of Natural History (negative no.338012). Earth-moon diagram, courrsy oflohn Michel. Mithns bu$ting fmm eg& photograph courtesy of the Musum of Antiquities of the University and Socity of Antiquaries of Newcasde upon Tyne. Statue of Athena. photograph courtesy of the Acropolis Museum, rlieI inv. no. 1 o/TAP SERITCE. Persephoneand llades, photo$aph courtsy of The British Museum (negarive no. PS259676). Sevn-twelve geomekic dia8ram, courtesy of John Michll. Birth ofvmus, photogaph courtesy of Galeria degli Ufiizi-Firnz. Native American maslg photograph couftsy oI the National Museum of the Amrican Indian, Smithsonian Institution. Tripartite seffdrawing, couftsy of the School of the Narural Order, Bakr, Nevada. Pruvian nose .ing, photograph coutsy of The Metropolitan Museum of Art, The Michael C. Rockefeller Memorial Collecfio& Bqu6rof NelsonA. Rnkefe]let 1979. 09n.206.1772) TunS dynasty mirror back, photoFaph ourtesy of the Board of rrusres oI rhe victoda & A]bert Museum. Shaded tenfold geomerry design, courtsy ofjohn Michell. Pandise tuomriis Riches lle!/es nanuscript, photoSraph courtesy Instirut d France,Muse conds chareau de Chanfilly Cathedral facade drawing, courtesy of rhe Catlldral of Sr. Iohn the Divine, New york City.

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