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mathematical constants in nature, art & beyond


Š 2011 Kristin Riger. All rights reserved. The Academy of Art University San Francisco, California Content, design, photographs and illustrations by Kristin Riger. Special thanks to Kyprianos Elisseou, Javier Martinez Avedillo, Philipp Klinger and Simon Bradford for the use of their photographs. All other images copyright of Kristin Riger. Taurus Bookbindery San Francisco, California www.taurusbookbindery.com 415.671.2233


mathematical constants in nature, art & beyond BY KRISTIN RIGER


To see a world in a grain of sand, and heaven in a wild flower, hold infinity in the palm of your hand, and eternity in an hour. — William Blake

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  Sacred Geometry of the Ancient World


TABLE of CONTENTS 1. Ancient Origins of Sacred Geometry

6-35

Basic Concepts

10-19

Ancient Architecture

22-35

2. The Geometry of Nature’s Harmony

36-71

Natural Forms

40-61

The Human Body

64-71

3. Sacred Geometry in Modern Culture

72-91

Modern Architecture

76-87

Art & Music

88-91

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1

ANCIENT ORIGINS OF

SACRED GEOMETRY


1

the origins of

SACRED GEOMETRY By definition, geometry is the archetypal patterning of all things, be they noumenal (something whose experience may be felt but not proved), conceptual, mathematical, natural or architectural. Geometry was considered sacred because it was the most concrete, as well as the most abstract, form of reasoning. Almost all ancient civilizations created their sacred monuments with careful reference to the correct numbers, geometry and proportion. Geometry governed the very movement of the heavenly bodies and the seasons. Subsequently, geometric concepts were applied to the positioning and orientation of their constructions. The ancient Greeks were the first civilization to establish the study of sacred numbers and geometry, although they may have learned the basics from ancient Egypt. This knowledge was preserved by the Arab world during the early Middle Ages, and eventually returned to Western Europe in the 12th century with the appearance of translations of Arabic and Greek texts into Latin. Geometry provided the Greeks with absolute truths that could be proved over and over again with the simplest of tools­ — a compass and a straight edged ruler. The subtleties of number and absoluteness of geometry were a part of the noumenal world, the hidden structure behind physical matter. They conceived the creator

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  Sacred Geometry of the Ancient World


of the universe in terms of absolute truth, not in terms of handed-down religious belief. They deduced that numbers were essential to the universe and that creation proceeded from abstract forms. Geometry, in its purest, simplest form, is sacred. Yet it’s founded on ordinary geometry and basic geometric figures such as circles, triangles and squares, as well as ratios and harmonics. Just as growth is expressed by repeating patterns, so art and virtuosity in architecture are often expressed by harmony.

Geometry is considered sacred because it codifies the order behind creation. It is the instrument used to create the physical universe. Classical Greek philosopher and mathematician Plato believed that all things grew from simple geometric forms. Initially, his ideas were considered mystical, but the physical importance of simple forms and numbers is now being confirmed by physicists and biologists who have discovered essentially simple formulae, such as the structure of DNA (based on the geometry of the double helix and the pentagon) and the pattern of leaf growth in plants (based on a fixed geometric angle). In the Middle Ages, the university curriculum was called the trivium, which focused on grammar, rhetoric and logic. The more advanced course, however, was called the quadrivium (literally ‘four subjects’) and reflected the importance of geometry, arithmetic, astronomy and music. These are all connected by the geometry of the Greek mathematician Euclid, and the numbers of Pythagoras (sixth century b.c). Music was seen as a matter of arithmetic — the precise arithmetic divisions between adjacent musical notes defined harmony and so formed an arithmetic you could hear. Sacred proportions are governed by numbers, known as phi (Ø), or the golden mean or golden section. They occur continuously in the works of the ancient Greeks, in Medieval architecture, and the growth of living things. Through these numbers, the geometry of nature and the perspectives of art and architecture coincide.

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BASIC CONCEPTS OF SACRED GEOMETRY

PLATONIC SOLIDS The platonic solids, the basic shapes of sacred geometry, are five three-dimensional geometric forms of which all faces are alike, and each platonic solid represents one of the five elements of creation. Plato called these 3-D polygons perfect, and he defined the five solids as follows: Tetrahedron = 4-sided (tetra = 4) Hexahedron/cube = 6-sided (hexa = 6) Octahedron = 8-sided (octa = 8) Dodecahedron = 12-sided (dodeca = 12) Icosahedron = 20-sided (icosa = 20) These shapes were considered special because each formation has the same shape on every side, every line on each of the formations is the same length, every internal angle on each of the formations is exactly the same, and each shape fits perfectly inside a sphere, all points touching the edges of the sphere. Each shape can also be attached to the same shape or other platonic shape to generate a bigger platonic solid. An interesting characteristic of the five platonic solids is their “duality”. The dual of a platonic is the shape formed with its vertices at the centre of each face of the parent platonic. The tetrahedron is the dual of itself, whilst an octahedron is the dual of a hexahedron and vice versa, Icosahedron = Water

and a dodecahedron is the dual of the icosahedron and vice versa. Thus, each platonic can have nested platonics within it of diminishing sizes down to an infinetely small side lengths, and yet every nested structure will still have all characteristics of a platonic solid. In the case of the tetrahedron, where the number of faces is equal to the number of vertices, its dual will be the same shape of its parent platonic shape.

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  Sacred Geometry of the Ancient World


Tetrahedron = Fire

ELEMENTS of CREATION

Octahedron = Air

The Ancient Greeks called the five solids the atoms of the Universe. In the same way that we believe that all matter is made up of combinations of atoms today, so the Greeks believed that all physical matter is made up of the atoms of the Platonic solids and that all matter also has a mystical side represented by their connection with earth, air, fire, water and ether. Indeed, the mechanism of platonic solids is so perfect, that perhaps their concept of platonics as being the building blocks of matter might be more evolved than our present knowledge of the atom model.

Hexahedron = Earth

Dodecahedron = Universe

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SYMBOLS of LIFE In addition to the Platonic solids, there are several geometric figures that are symbolic to the concepts of sacred geometry (pictured right). Metatron’s Cube is a two-dimensional figure created from 13 circles with lines from the center of each circle extending out to the centers of the other 12 circles. Six circles are placed in a hexagonal pattern around a circle, with six more extending out along the same radial lines. The Flower of Life is composed of evenly-spaced, overlapping circles that form a pattern with a sixfold symmetry. The Flower of Life is said to contain sacred value as a visual expression of the connections life weaves through all beings. Depictions of the Platonic solids are found within the symbol of Metatron’s Cube, which is derived from the Flower of Life pattern, in addition to several others: 1. The Fruit of Life is said to be the blueprint of the universe, containing the basis for the design of every atom, molecular structure and life form. 2. The Seed of Life is formed from seven circles being placed with sixfold symmetry and is a symbol depicting the seven days of creation. 3. The Tree of Life is a concept, a metaphor for common descent, as well as a motif in various world theologies and philosophies. This has historically been adopted by some Christians, Jews, Hermeticists, and pagans.

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M E TR AT RO N’S CUBE

FLOWER O F LI FE

FRU IT OF LIFE

SE E D O F LI FE

TREE O F LI FE

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the FIBONACCI SEQUENCE The Fibonacci sequence is a series of numbers in which each number in the sequence is the sum of the two preceding numbers, 0, 1, 1, 2, 3, 5, 8, 21, 34, 55, 89, 144, and so on. Each term in the Fibonacci sequence is a function of its preceding values. Patterns exhibiting the sequence are commonly found in natural forms, such as the petals of flowers, spirals of galaxies and bones in the human hand. Its ubiquity has led many to conclude that its shape is intrinsically aesthetic, making it one of the most influential patterns in design and mathematics in human history. The Fibonacci series was discovered by the Egyptians and the Greeks. Edouard Lucas in the 19th century named the series after Leonardo of Pisa, also known as Fibonacci, who made the series famous through his solution of a problem regarding the breeding of rabbits over a year’s time. Fibonacci sequences are most often used in conjunction with the golden ratio. Two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874. Other names frequently used for the golden ratio are the golden section and golden mean. The diviThe Nautilus (left) The nautilus shape is an

sion of any number in the sequence by an adjacent number yields an approximation of the golden ratio.

example of the Fibonacci sequence found in nature.

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9

1

36 8

1 1

28 7

56

1

126 70

15 5

10

1

56

20

1

2 1

1

the FIBONACCI FORMULA By definition, the first two numbers in the Fibbonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two values. Each further term of the sequence is defined as a function of the preceding terms, and as a result, the equation recursively defines the sequence once one or more initial terms are given. The diagram above illustrates how the sum of preceding numbers affect the following values in the Fibonacci sequence.

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation: Fn = Fn-1 + Fn-2 Fibonacci numbers can be found in various ways in the sequence of binary strings. The number of binary strings of length n without two consecutive 1s is the Fibonacci number Fn+2. By symmetry, the number of strings of length n without consecutive 0s is also Fn+2. The number of binary strings of length n without an odd number of

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  Sacred Geometry of the Ancient World

1 1

1 1

1 1

1 1

1

consecutive 1s is the Fibonacci number Fn+1.

5

3

8

6

4

9

7

21

10

3

36 28

15

6

4

1

84

35

35

21 6

1

126

84


PAS CA L’S T R I A NG L E Pascal’s triangle is an array of the binomial coefficients in a triangle and is named after the French mathematician, Blaise Pascal. It is known as Pascal’s triangle in the Western world, however, mathematicians have studied it centuries before him in India, Persia, China, Germany, and Italy. The rows of Pascal’s triangle are staggered relative to the numbers in the adjacent rows. In 1068, a mathematician Bhattotpala realized that the shallow diagonals of the triangle sum to the Fibonacci numbers, as shown in the diagram below.

1

1

2 1

3

1

5 1

8

2

8

1

13

13 1

21

3

3

1

21

34

34

55

1

4

4

6

1

55

89

89

144

1 6

1 1 1 1 1

7 8

9 10

15 21

28 36

45

10

5

20 35

56

210

5 15

35 70

126

84 120

10

6 21

56 126

256

1 1 7 28

84 210

144

1 8 9

36 120

1

45

1 10

1

17


Construct your golden rectangle by tiling squares which are Fibonacci numbers in length, 1, 2, 3, 5, 8, 13, 21, and 34. This will form the basis for the golden spiral.

1. Construct a square.

2. Extend a line to the right that is half of the width of the square, or the length of b.

1

18 

b

  Sacred Geometry of the Ancient World

the length of the square to close the shape.

3

2 a

3. Construct a parallel and perpendicular line

b

c


4. Place squares the length of

5. Continue to divide the shape

c and d, to the top and left.

into smaller Golden Rectangles.

6. Construct the Golden Spiral through the Golden Rectangle.

d

4

5

6

c

When a square section is removed, the remainder is another golden rectangle with the same proportions as the first. Corresponding corners of the squares form an infinite sequence of points on the golden spiral.

19


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  Sacred Geometry of the Ancient World


21


SACRED GEOMETRY IN AN CIENT ARCHITECTURE

GREAT PYRAMIDS OF EGYPT The subject of pyramids makes people think of the Great Pyramid and its immediate neighbors. In fact, the Great Pyramid is simply the largest of a succession of more than 35 pyramids, stretching over a long period of Egyptian history. Egyptians designed all the pyramids using a calculation called the “seked,” the measure of gradient or inclination of any one of its four triangular faces to the horizontal plane of its base. The seked was mostly expressed in horizontal palms per one vertical cubit rise — in modern geometric terms, this means that the cotangent of the angular slope of the triangular faces. The seked can be a range of values in any given Egyptian pyramid, but two groups clearly stand out: those pyramids with a seked of 5.25 and those with a seked of exactly 5.5. Typically, pyramids with a similar seked are geographically close to one another. The seked measurement is neither arbitrary nor approximate. The distance around the base of the Great Pyramid’s base exactly equals the circumference of a circle who radius is the height of the pyramid.

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  Sacred Geometry of the Ancient World


The illustration above shows how the Egyptians calculated the seked in order to construct their pyramids by drawing the right-angled triangle. The distance FE is the same as half the base length, or half DA.

T H E G O L DE N MEA N IN THE G REAT PYRA MID It is clear that Egyptian architects used careful considerations of the Golden Mean when constructing the Great Pyramid. By choosing the proper dimensions, they ensured that the area of each face of the Great Pyramid is exactly equal to the square of its height. This is accomplished by choosing the slope of the pyramid such that the apothem equals 1.618 times half the side of the base. In the early descriptions of the pyramids, the measurements were in cubits, where the Great Pyramid’s apothem was 356 cubits and the base, 440 cubits. If one divides 356 cubits by half the base, 220 cubits, we yet again find the Golden Mean. This is found by dividing 356 and 220 by four, in order to arrive at the ratio of 89 divided by 55. Both being Fibonacci Numbers, the answer is 1.618. Finally, when the Great Pyramid is viewed from the side, the laws of perspective reduce the area of the face to the correct size for the projection, which equal the Pyramid’s cross section. Thus, the viewer sees is the correct triangle.

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the ATHENIAN PARTHENON The Parthenon is a sacred precinct built by the minds of the Ancient Greek culture that invented geometry. It is the archetypal example of sacred geometry and is known as a geometrical masterpiece. It is very pleasing to the eye due to the unique architecture. The lines that are perceived as horizontals in fact are curved upward in the middle. The platform upon which the columns of the temple stand is slightly curved on all its four sides. Each of the columns are tilted inward slightly, and are placed closer together toward the corners of the ancient building. The Parthenon was designed in such a way that it conveys several sacred geometric proportions, and the use of phi is particularly prominent in the proportions delineated by its facade. Specifically, if the height of the structure from the base to the peak is taken to equal one, then the width of the Parthenon across the facade measures exactly 1.618. The placement of the pillars, and many other aspects of the building, are then determined by exponential factors of phi, such as one divided by phi over two, three, etc. The placement of phi in the Parthenon is noted in the illustration above.

Columns (right) Two of the Parthenon’s eight columns on the front facade. Eight, of course, is a Fibonacci number.

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  Sacred Geometry of the Ancient World


C H ARTR ES CATHE DR A L The French medieval Cathedral of Our Lady of Chartres is considered one of the finest examples of the French High Gothic style. The majority of Chartres’ original stained glass windows survive intact, while the architecture has seen only minor changes since the early 13th century. The building’s exterior is dominated by heavy flying buttresses which allowed the architects to increase the window size significantly, while the west end is dominated by two contrasting spires. Equally notable are the three facades, each adorned with hundreds of sculpted figures illustrating key theological themes. The growth of Gothic architecture might have derived from the intense interest in the proportions of the original Temple of Solomon, which appeared to showcase an extraordinarily high roof. Chartres provided the roots of many great Gothic cathedrals, specifically the design of the flying buttress. The spires of Chartres Cathedral, which symbolize the sun and the moon, are also the most obvious of of the evident geometric symbolism embedded in its designs.

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  Sacred Geometry of the Ancient World


The geometry of the flying buttresses had to be calculated very carefully to transfer the huge weight of the roof span downward to the ground rather than outward, which would have destroyed the walls. This geometry is based on circles. The center of the circles in the buttress cross-section fall on the same vertical line drawn down the face of the wall. These three circle centers also correspond to the three main levels. There have been many attempts to interpret the sacred geometry of Chartres Cathedral. Gothic architecture scholar John James applied a series of geometrical figures to the plan. He claimed that the Middle Ages favored multiplicity as part of the divine order, and that there was several geometric systems in the cathedral, while being locked together at a few essential points like the labyrinth and the altar, the most sacred in the building.

Sacred architecture aims at eternity; and therefore, is the only thing incapable of modes and fashions in its principles. — Sir Christopher Wren The direction the cathedral faces is noteworthy of mentioning. While most Christian cathedrals face east, Chartres faces northeast. The cathedral is aligned to the summer solstice. On the summer solstice, the Sun shines through a window named ‘Saint Apollinaire’ with a depiction of the Roman sun god Apollo and its rays fall straight on an iron nail in the floor of the cathedral. Sacred geometry, of course, is needed to locate and align such sacred structures.

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C ONSTRUCTI NG C H A RT R E S ’ NO RT H RO S E WIN DOW

The illustration below shows a basic geometric construction of the North Rose window at Chartres Cathedral. Using a circle, place twelve equally-spaced points around the outside. Connect every fifth point to create a dodecagram, or a twelve-pointed star. Twelve small circles nestle within the rays of the star. Find the outer point of these circles, and connect them to form a smaller dodecagram star.

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  Sacred Geometry of the Ancient World


After both dodecagrams have been constructed, erase some of the inner line segments, and connect points with twelve central lines as shown. Once the inner star has been filled in, this is enough to reveal the basic scheme of how this rose window was designed. The geometry of this window is an integral part of the design of the cathedral which was composed as an interconnected whole.

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31


ANC I ENT L ABY R IN T H S Labyrinths can be found in almost every religious tradition around the world. During the Middle Ages, labyrinths were strategic geometric patterns that related to the whole the cathedral. The concept of unity, symbolized by the circle, stands behind the choice of numbers, proportions and pattern.

The labyrinth is a symbol for life’s journey  — The labyrinth path, representing creation, direction and life’s choices, winds into the center winds in a clockwise pattern, and the path unwinds counterclockwise, representing the different paths that we take in life. The spiral is a basic form of nature, although the labyrinth spiral is more complex. The circular path inward cleanses and quiets the mind. The unwinding path integrates and empowers on the walk back out. Ancient labyrinths were most often paths of eleven concentric circles with a twelfth being the center of the labyrinth. Twelve was an important number in the sacred arts, as it is the multiple of three, representing heaven, and four, representing earth. The labyrinth’s center is made up of a six-petaled rose-shaped area called a rosette. The rosette was a sign of beauty and love that dates as far back as the Egyptians. The rose is regarded as a symbol of enlightenment even today.

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  Sacred Geometry of the Ancient World


Chartres Labyrinth (above) The design for the labyrinth in Chartres Cathedral in France. A replica appears of the floor of the Grace Cathedral in San Francisco.

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  Sacred Geometry of the Ancient World


35


21

2

THE GEOMETRY OF

NATURE’S HARMONY 13 3 5

1

2 8

34


2

the geometry of

NATURE’S HARMONY The numbers and shapes of sacred geometry used by the Greeks are reflected in a vast array of natural forms. In plant life, the easiest numbers to measure are those of seeds in a flower or the angles at which successive leaves grow from a central stem. Both these occurrences follow the Fibonacci series. Documenting the mathematics of the living growth is often difficult, but it can be seen in the forms of shells or horns, which are concrete traces of growth. In the nautilus and in fossil ammonites, we can see the geometry of successive chambers adhering closely to the geometry of the logarithmic spiral. In the horns of animals, we can see similar spirals governed by other geometric formulae. In the mineral world, the large variety of crystals utilizes only seven different geometric forms. The structure, properties and qualities of that universal solvent of life, water, in both its liquid and snow forms, subscribe to the same geometry. Finally, at the molecular level lies the subtle geometry of DNA — a double helix surrounding a double pentagonal structure. Truly, the geometry of nature is sacred.

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  The Geometry of Nature’s Harmony


Because sacred geometry forms the basis for the construction of the universe, we might expect to find it also in the design of both animate and inanimate natural things. For all nature’s apparent richness, the pattern is amazingly complex, but it is based on simple building blocks. In 1753, Scottish botanist Robert Simson realized that Fibonacci’s series governed the growth pattern of many plants. Essentially, it maps the geometry of the growth —  the equiangular spiral and phi are also found in the spacing of leaves on a stem, in petal numbers and in the arrangement of seed heads.

The natural growth of all living things consists of replicating patterns. A plant produces leaves that conform to the pattern inherent in its species — they grow out from stems at geometrically predictable intervals. Look down on a straight plant stem and you will see that the leaves protrude from the stem in a spiral pattern. This spiraling gives each leaf or branch access to the maximum amount of sun or rain. By tracing your finger from leaf to leaf around the stem, you can establish the order of growth — it takes the shape of a helix. You can also establish two numbers: the number of leaves protruding from the stem and the number of rotations your finger moved around the stem in order to count the leaves. The clearest examples of the presence of Fibonacci numbers in nature are pine cones and sunflowers. The sunflower spirals, one left-handed and one right-handed. There may be eight right-handed spirals and 13 left-handed spirals, each seed belonging to both. Other pairs include 34 and 55, or 55 and 89. The Fibonacci series also seems to determine how many petals a plant will have on its flowers. The number of petals never reaches 144, a number that is often found to be limiting in other examples of the Fibonacci series in nature.

39


GEOMETRY IN NATURAL FORMS

GROWTH and DIMINUTION Nature pulses with cycles and rhythms of increase and decrease. Observe the breath of the tides, the waxing and waning of the moon, the circle of the year, the interplay of day and night, the beat of the heart, and the expansion and contraction of the lungs. The explosion of a star is often followed by implosion, and the negative entropy in the ordered organization of life is balanced by the positive entropy of disorder and death. In chaos theory, the golden section governs the chaos border, where order passes into and emerges out of disorder. Demanding simplicity and economy, nature appears to require an accretion and diminution process that is simultaneously additive and multiplicative, subtractive and divisional. This demand is satisfied perfectly only by the golden section powers, and in practice by Fibonacci and Lucas approximations. Think of an oak tree. It shoots up as fast as it can from an acorn, only to slow, mature, fractalize its space toward a limit, becoming a new relative unity. Nature simultaneously grows and diminishes to relative limits. Recognizing that structural self-similarity connects, or binds, what he called the hidden “implicate order” to the outer “explicate order,” physicist David Bohm remarked: “The essential feature of quantum interconnectedness is that the whole universe is enfolded in everything, and that each thing is enfolded in the whole.”

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  The Geometry of Nature’s Harmony


O R DER i n DIV ERSITY Many living things, such as plants, trees, insects, fish, dogs, cats, horses and peacocks, each display a poetic interplay between symmetry and asymmetry in their growth patterns. The prevalence of natural pentagonal forms, for instance, may result from the symphony of golden relationships in the pentagon and pentagram. Many marine animals, like starfish, exhibit 5-fold form. Sometimes, as in a passion flower, the form is decagonal, one pentagon superimposed upon another. In any case, nature exhibits some form of pattern in the growth of almost every living things. Even the building blocks for life, ammonia (NH3), methane Growth Patterns (above)

(CH4), and water (H20) all have internal bond angles which

The plant pictured displays symmetric

approximate the internal 108째 angle of a pentagon.

and asymmetric growth patterns.

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  The Geometry of Nature’s Harmony


GEOMETRY in SEED GROWTH The grouping of seeds in a plant’s seed pod or base is often directly determined by the Golden Section. One of the most beautiful and well known examples of the use of phi in seed organization is in the sunflower. In the seed base located in the center of the sunflower, the seeds are laid out according to a specific geometric pattern. If closely observed, one can see that there is a pattern involving two sets of spirals that intersect, with one turning clockwise, and the other turning counter-clockwise. At each location where two spirals cross a seed can be found. The number of spirals turning counter-clockwise happens to be 55, and the number of clockwise spirals happens to be 89 — both Fibonacci numbers. If the rotational rates of these two sets of spirals are analyzed, it is evident that they are ruled by the Fibonacci spiral discussed in the last section.

Flower Seeds (left) A flower seed is a well known example of the Golden Section seen in plant growth.

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137.5°

As the seed head expands, new seeds are formed at angles of 137.5°.

S E E D PAT T E R N S ( P H Y L L O TAX I S ) Emerging as a science in the 19th century, phyllotaxis is the spiral growth pattern of plant seeds, such as a sunflower head, scales of pine cones and cacti. Da Vinci observed that the spacing of leaves was spiral in arrangement in the 15th century. It was later discovered that the majority of wild flowers are pentagonal, and that Fibonacci numbers occur in leaf arrangement.

The diagram shows the main features of spiral phyllotaxis. The growth, exhibits successive rotations with a constant separation distance, while distances in a logarithmic spiral form a geometric progression. In 1754, Charles Bonnet coined the name phyllotaxis from the Greek phullon “leaf” and “taxis arrangement.” The concept of the divergence angle, or the genetic spiral, which noticed the presence of simple Fibonacci numbers, was developed in 1830. In 1837, the Bravais brothers discovered the crystal lattice and the ideal divergence angle of phyllo-taxis: 137° = 360° / Ø2.

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  The Geometry of Nature’s Harmony


The curvature of the phyllotaxis, or seed growth, starts to form based on the logarithmic spiral.

The logarithmic spiral begins to connect the seed growth.

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S PI R AL I NG G ROW T H of a F L OW E R S E E D

The growth of a flower seed consists of replicating patterns. The pattern at the center of a daisy is shown in the illustrations below. The florets that make up this pattern grow at the meeting points of two sets of spirals, which move in opposite directions, one clockwise, the other counterclockwise. The concentric circles of the seed growth also grow according to the Fibonacci sequence, as each radius is the sum of the two before it.

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  The Geometry of Nature’s Harmony


Here two of the spirals have been reconstructed with the help of a series of concentric circles, at distances growing along a logarithmic scale, and a series of straight lines radiating from the center. If we connect the consecutive meeting points of these two sets of opposing lines, we can see the flower’s growth spirals. These spirals are logarithmic and equiangular, since the angle they describe with the radii remains the same.

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  The Geometry of Nature’s Harmony


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SPIRALS in LEAF GROWTH In nature, spiraling growth occurs through a simple accretive process. It produces the logarithmic spiral that we see in plants, which grow in size, increasing in length and width, without varying in proportions. As the central stem of a plant grows upward, leaves or branches sprout off of the stem in a spiraling pattern. In an over-simplified example, the plant grows up an inch, and a leaf or branch sprouts out of the stem. Then the plant grows up another inch, and once again a leaf or branch sprouts out, but this time it sprouts out in a different direction than the first. When connecting the tips of the leaves or branches that have grown out of the stem, it is apparent that a spiral patterns forms around the central stem. The diagrams pictured on this page illustrate this spiraling growth pattern. When a spiral is logarithmic, the curve appears the same at every scale, and any line drawn from the center meets any part of the spiral at exactly the same angle for that spiral. Zoom in on a logarithmic spiral, and you will discover another spiral.

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  The Geometry of Nature’s Harmony


FI BONAC C I L E AV E S Nature uses a variety of logarithmic spirals in leaf shapes. The angles between each leaf can be determined by dividing the number of turns (each turn 360 degrees) by the number of leaves sprouted over the distance. By counting leaf growth on the stems of plants such as an elm, cherry or pear, you will find a Fibonacci number. Leonardo of Pisa loved to observe and draw living structures, trees waves, anatomy and water flow.

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Despite its seemingly endless variety and diversity, nature employs three basic ways to arrange seeds or leaves along a stem: disticious, like corn, decussate, like whorled, such as mint, and the most common, spiral phyllotaxis, for about 80% of the 250,000 different species of higher plants. As the plant grows, leaves typically appear arranged around the stem in such a way that optimizes yield of light. Essentially, leaves form a helix pattern centered around the stem, which is both clockwise or counterclockwise, and depending upon the species, the same angle of divergence.

This opposing growth pattern aids photosynthesis, as each seed, or

OPPO S I NG S EED S

leaf, receives maximum sunlight and rain, efficiently spirals moisture to roots, and gives best exposure for insect pollination. Spirals of seeds in a sunflower, for example, often appear as adjacent Fibonacci numbers, such as 55:34 (1.6176) or 89:55 (1.6181). Pineapples have three spirals, often eight, thirteen and twenty-one each, where 21:13:8 approximates phi:1:1/phi, with 21:13 (1.6153), 13:8 (1.625) and 21:8 (2.625). Scales of pinecones are typically 5:3 (1.6176) or 89:55 (1.6). Artichokes likewise display eight spirals one way, five the other. The opposing spirals of seeds observed on the following page are an optical illusion due to the fact that the ratio of the successive Fibonacci numbers approximates phi. If you count the number of arms in these spirals, you will find they always equal two adjacent Fibonacci numbers whose ratios to the succeeding numbers are slightly above and below phi.

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  The Geometry of Nature’s Harmony


The following images exemplify how opposing spirals of seeds work together in a pineapple and an artichoke. Each plant shown has both steep and gradual spirals that diverge either clockwise or counterclockwise. While evolution produces as many seeds as possible, the Fibonacci sequence provides the mechanism for packing them together in the most efficient way.

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  The Geometry of Nature’s Harmony


55


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  The Geometry of Nature’s Harmony


WORLD of FRACTALS In 1975, mathematician Benoit Mandelbrot defined fractals as objects that do not lose their detail or their proportions when they are magnified or shrunk to the microscopic level. This property is highly reminiscent of the Golden Section, or 1.618, where the same essential and sacred proportion is retained every time you cut a line or rectangle. The qualities of both fractals and phi are concerned with growth. Fractals are popularly supposed to be associated with the mathematics of chaos, but they are, in fact, very ordered — millions of interlocking, self-replicating, natural objects. They only look chaotic, yet they are governed by a definite geometry. An example is the movement of clouds, which are definitely fractal controlled by the inherent properties of the interaction of water vapour with air and dust particles. The essence of measuring fractals is to isolate the basic pattern — what is called its initial recursive mathematical function. Interestingly, the Fibonacci series is one such recursive function.

Fractal Fern (left) A fern leaf is a common example of fractal growth in nature.

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Fractals in Nature (above) Fractal images pictured from left to right: cauliflower, red mulberry, pinecone and an elm.

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  The Geometry of Nature’s Harmony


FRACTAL FEATURES Plants, often understood as chaos in action, actually follow subtle fractal laws of nature. The fractal law of self-similarity is best observed through the branching process of plants — the main stem or trunk splits into a number of branches, each splitting into smaller branches and so on. L-systems, developed by Lindenmayer, have been used to model the fractal branching of plants. Fractals can be observed in many plants (ferns, cacti, trees) and also in vegetables like cauliflower. The photos below are of naturally occurring fractals observed in plants. Another important feature of fractals is scaling. The degree of fragmentation is identical at all scales. Fractals do not become smoother as the magnification brings you closer; they continue to generate new irregularities.

Fractal geometry can lead to convincing images of natural growth phenomena, such as coastlines, ferns and tree bark. They can also emerge from climate and even manmade phenomena, such as stock price graphs or economic predictions, which show self-similarity. The snowflake is an example of a geometric fractal that grows by the addition of equilateral triangles in specific patterns. Some ferns are classic natural examples of a fractal pattern, with section of the leaf being a miniature replication of the whole leaf. This indicates that nature does not have to redesign the leaf at every stage of its growth, but the initial design just keeps on replicating.

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Place 6 stars with adjacent points to form a 6-sided shape, and then remove the interior lines.

Construct a star with two overlapping triangles.

FRACTAL GEOMETRY in SNOWFLAKES The structure of a snowflake is one of the clearest manifestations of fractal patterns seen in nature. This may be because snowflakes form when water falls through the atmosphere without interference from adjacent objects. No other substance crystallizes in so many ways. Despite such variety, the geometry that governs the growth of one of a snowflake’s branches will govern the growth of its other branches. The procedure of generating the fractal representation of a snowflake may not reflect what actually happens structurally on a freezing cold day, but it gives a fair mathematical representation of the fractal nature of a snowflake. It also demonstrates that, given a sharp enough pencil and superhuman eyesight, the process could go on till you reach the infinitely small. With fractal geometry, the length of the perimeter of the figure increases without limit, but length of the perimeter of the snowflake depends on the degree of magnification that is used. It can also be shown mathematically that the area of the snowflake will never exceed 1.6 times the area of the original generating triangle, which is, again, a Fibonacci number.

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  The Geometry of Nature’s Harmony


Repeat this exercise to continue the replicating pattern.

Notice the infinite nature of this exercise, which exemplifies the fractal qualities of the snowflake.

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PHI IN THE HUMAN BODY

THE VITRUVIAN MAN The perception of human proportions has varied greatly throughout the ages. One of the earliest written documents dealing with human proportion is by Marcus Vitruvius Pollio, the first-century Roman architect and writer. He begins his book Ten Books in Architecture with the recommendation that temples should be constructed on the analogy of the wellshaped human body, in which, he says, that there is a perfect harmony between all parts. Vitruvius described the human figure as being the principal source of proportion among the classical orders of architecture. Vitruvius also suggests that the height of a proportional man is the same as the span of his outstretched arms. These two equal measures yield a square which encompasses the whole body, while the hands and feet touch a circle centered upon a navel. The Vitruvian Man is a world-renowned drawing created by Leonardo da Vinci in 1487 based on the correlations of ideal human proportions with geometry as described in Vitruvius’s book. Leonardo, who studied harmonious proportions and the golden section throughout his life, summarized his findings of good proportion in memorable words:

For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square. — Leonardo da Vinci

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  The Geometry of Nature’s Harmony


SQ UA RING t he C IRC L E The relatedness of the human body to the circle and to the square rests upon the archetypal idea of squaring the circle, which fascinated the early discoverers of the phenomenon, because these shapes were considered perfect and even sacred, the circle having been looked upon as a symbol of the heavenly orbits, the square as a representation of the four squares of the earth. The two combined in the human body suggests in the language of symbolic patterns that we unite within ourselves the diversities of heaven and the earth, an idea shared by many mythologies and religions. The drawing of a human figure to the left illustrates raised arms and outstretched legs that touch yet another circle, Vitruvius Man (above)

beyond the one centered at the navel. The unity we share

A basic diagram of Leonardo’s

with plants and animals is apparent from the fact that our

famous drawing.

growth, like theirs, seems to unfold from a single center, which in our case is at the top of the sacrum. It may be recalled how the spirals of the daisy and sunflower also unfold from the center.

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LE COR BUS I E R ’S MO DU L O R

b

c a

French architect Le Corbusier devised a modern version of the golden section using a scale of proportions developed from the length of the human body seen here. He termed the system Modulor, and saw it as a universal tool for determining proportions for architecture and design. b c

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  The Geometry of Nature’s Harmony


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8

5

3

2

SAC R E D BODY PRO P O RT I O NS Many cultures and spiritual traditions have always sought to discover what defines human perfection. We are heirs of many visions and concepts relating to the harmony of the human being. The human body is an example of nature’s proportion. Phi, the golden number 1.618, is a proportion found in many areas of the natural world as well as in the structure of the human body. The bones that make up our skeleton are thought to have a proportional relationship of 1:1.618.

More studies are being done in finding the golden ratio in the human hand. In the example above, you’ll see how Fibonacci numbers can be found by studying the bone structure of the hand. But the hand is only one of many occurrences of the golden ratio in our anatomy. The human face abounds with examples of the golden section. The head forms a golden rectangle with the eyes at its midpoint.The mouth and nose are each placed at golden sections of the distance between the eyes and the bottom of the chin. Phi defines the dimensions of the human profile. Even when viewed from the side, the human head illustrates the golden proportion.

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  The Geometry of Nature’s Harmony


L E NG T H o f t he H U MA N BO DY The golden section is manifested in both the structure and length of the human body. If the length of the hand (e) has the value of 1, for instance, then the combined length of the hand and the forearm (c) has the approximate value of phi.

a b

The distance from the head to the shoulder blades (b) is a golden section of the distance between the skull and

d

the navel (d) yellow line. This pattern continues with the distance between the head and the fingertips (f), which

f

is a golden section of (d). These relationships relating to phi occur throughout the entire body. The human body is based on phi and the number 5, which

g

is a Fibonacci number. The number 5 appendages to the torso, in the arms, leg and head. 5 appendages on each of these, in the fingers and toes and 5 openings on the face.

c e

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DO U B L E H E L I X Deoxyribonucleic acid, or DNA for short, is composed of two right-handed 3-D helices. DNA combines into strands called chromosomes. Different species have a different number of chromosomes: humans have 46, or 23 pairs. DNA is like a spiral ladder with a series of rungs holding together the two strands. The double helix requires ten rungs to make a complete turn — the Kabbalistic Tree of Life has a ladder of ten rungs, and ten was Pythagoras’ number of completion.

The construction of the DNA structure uses a simple modular pattern. The rungs of the DNA ladder are composed of four types of molecules which are called nucleotides. Like the logarithmic spiral, the geometry is easily replicated (but better packed), and its facility for self-replication and growth is built into the DNA molecule’s geometry. The geometry of the DNA base is characterized by six coordinates that define the orientation of the base in a nucleic acid molecule relative to its predecessor along the axis of the helix. Together, they characterize the helical structure of the molecule.

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3

SACRED GEOMETRY IN

MODERN CULTURE


3

sacred geometry in

MODERN CULTURE Over the last several years, sacred geometry has become a topic of interest again within esoteric circles. As we know now, its roots can be traced not back to the Renaissance and the ancient classical world of ancient Greece, Egypt, and Persia. It is from the classic Greek writers, philosophers, and mathematicians that we in the modern world get the vast majority of our concepts for geometry and how it was and is still used in various arts and sciences. Sacred geometry was considered by the ancient Greeks as one of the most tangible forms of reasoning, and it was used not only for the solving of mathematical problems, but also for architecture, construction, and the art of sculpture. Modern day architects and builders are now using these ancient sacred concepts as inspiration for cutting edge architecture that uses structural shapes which have never been used before. At the heart of these new design concepts is a common affinity to building our surroundings in relation to the principles and patterns that govern nature that surrounds us.

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  Sacred Geometry in Modern Culture


In addition to architecture, the golden section is just as evident in creative communities today as it was in the Middle Ages. Musicians, designers, painters and photographers all still apply divine proportions to their work to create balance, harmony and visual interest.

The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty. — Wentworth Thompson Sacred geometry is not just evident in large scale buildngs or in artistic masterpieces, but in places or products that we see and use everyday. Today, we consistently find ourselves reaching for that plastic card in our wallets and handbags. Most credit cards measure 86mm by 54mm, almost exactly an 8:5 rectangle and one of the most common Fibonacci approximations to the golden rectangle. Because of its aesthetic qualities, embodied in its unique ability to relate the parts to whole, golden ratios are used in the design of many modern household items, from coffeepots, cassette tapes, playing cards, pens, radios, books, bicycles, and computer screens, to tables, chairs, windows and doorways. It even comes into literature, in the page layout of manuscripts and designs. There are other important rectangles also find their way into our daily lives. The geometric proportion is perfectly expressed in the golden series is mimicked in the International Standard Paper Size, which employs the continuos geometric proportion of 2:2:1. Whereas removing a square from a golden rectangle produces another golden rectangle, folding a rectangle in half produces two smaller rectangles.

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GEOMETRY IN MODERN ARCHITECTURE

SACRED STRUCTURES Sacred architecture doesn’t necessarily mean churches and temples, but of all buildings that are designed in accordance with the laws of sacred geometry. And as we have learned, sacred geometry is primarily concerned with the esoteric and symbolic significance of geometrical forms in conjunction with logical development. Buildings can have a profound influence on our health and spiritual state of being. Harmony and balance, light and color, relationship to landscape, ecological sympathy, energy efficiency and geometric form are contributing elements of shelter which aspires to be nurturing rather than draining. By creating an environment around us that is supportive to our senses, we can enhance rather than alienate our human links with nature. The architecture of the buildings we inhabit as our homes, our work places, our healing and education centers, and our churches and temples, has a similar impact on our senses as the architecture of nature. It is mainly the geometric spaces and the materials of the structures that influence our responses. Our sense of proportion tells us when a space or object is harmonious, and our mood responds accordingly. Modern architecture uses the laws of sacred geometry to create structures that are harmonious to the surrounding environment. While secular structures clearly had the greater influence on the development of ancient architecture, these concepts are being used in buildings for everyday purposes. Architects today incorporate sacred geometry concepts from the Greeks to create spaces that borrow from the past, but still look forward to the future.

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  Sacred Geometry in Modern Culture


S P I R A L STA I RC A SES Although it was believed that the first spiral staircase was used at least 3000 years ago in Sicily, the architectural structures remain popular even today, and there are a wide variety of examples and different variations of them which can be found in modern architecture. Spiral staircases give the illusion of a logarithmic spiral, which goes outwards from a fixed point. Some are structured as two separate helixes, one leading up and the other leading down, that twist together in a double helix formation. The following page showcases an example of a spiral staircase built in the 20th century in San Francisco at the Mechanics’ Institute Library. Other examples around the world are located in the Vatican Museum in Itlay, the Guggenheim in New York and the Sagrada Familia in Spain.

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  Sacred Geometry in Modern Culture


79


The desire to use geometric proportions found in nature for architectural design is both aesthetically gratifying and a realization that it’s not just temples that deserve this treatment, but buildings with other purposes as well. With the new study of the underlying mathematics of nature’s forms and the use of non-linear geometry and computer modelling, architects are now able to explore an exciting and littleknown world of non-linear organic structures. Nature is now viewed as being a mixture of order and chaos, pattern and accident, simplicity and complexity. It is not surprising that this has inspired new concepts in modern architecture.

NEW CONCEPTS

By bringing the curves of nature back into buildings, architects have reawakened the idea that a building can echo archetypal shapes rather than just being a cubic accommodation box. Modern architects have created a kind of fluid architecture that rejects the geometry of classical building and embraces organic and flowing forms. By bringing the curves of nature back into archit ectural structures, they have reawakened that a building can echo archteypal shapes rather than just being a cubic accommodation box. The first architect to anticipate the surrealists and achieve this on a grand scale was Spanish architect Antonio Gaudi, Sagrada Familia. Gaudi carefully observed natural forms and was a structural innovator. He used geometric models made of string and weights to predict the chain shape and stresses of his structures. The controversial concrete and ceramic roof of the Sydney Opera House, designed by Danish Jørn Utzon, is another example of modern architecture which echoes the shells of the nearby sea but all the essence of a complex geometric problem.

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  Sacred Geometry in Modern Culture


The following images show some famous examples of modern interpretations of sacred geometry. Top Left: The Sydney Opera House, Sydney. Top Right: Guggenheim Museum, New York. Bottom Left: The Louvre, Paris. Bottom Right: St. Mary’s Cathedral, San Francisco.

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  Sacred Geometry in Modern Culture


MODERN ORGANIC ARCHITECTURE The design for St. Mary’s Cathedral, by architect Paul Ryan, in San Francisco utilizes a hyperbolic paraboloid for the shape of its 200 foot high roof. A hyperbolic paraboloid is an infinite surface in three dimensions with hyperbolic (concave and convex curves) and parabolic (u-shaped curves) cross-sections. It can also be referred to as a saddle roof, which follows a convex curve about one axis and a concave curve about the other. The hyperbolic paraboloid form has been used for roofs in the past century since it is easily constructed from straight sections of lumber, steel, or other conventional materials, and forms a strong basis for extremely tall structures. This geometrical formula had not been discovered during the days of the great Gothic cathedrals — perhaps such computer-generated designs are part of the future of sacred geometry.

Hyperbolic Parabloid The roof of St. Mary’s Cathedral is designed around a hyperbolic parabloid, a new geometric shape.

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t h e SY D NEY OPER A H O US E The Sydney Opera House is a masterpiece of modern organic architecture and an iconic building of the 20th century. The massive concrete sculptural shells that form its roof appear like billowing sails filled by the winds. The Sydney Opera House was one of the first organic architectural buildings in the world to use computers in its design process. The opera house’s architectural form consists of three groups of interlocking shells, which are set on a vast terraced platform. The shells are faced in glazed off-white tiles while the podium is clad in earth-toned, reconstituted granite panels. Engineers could not calculate the geometry for the shell until the forces and bending movements were known, which took almost a year to determine. The influences and inspirations that resulted in the Sydney Opera House’s unique form include organic natural forms and an eclectic range of aesthetic cultural influences, unified in the one sculptural building.

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SACR E D I N F LU E NC E S In the Sydney Opera House, architect Jørn Utzon realized the great synthesis of earth and sky and landscape and city in terms of a unity of technological and organic form. As the geometry of the shells define the structure, so does the idea that structures inspired by nature are the future for design through the use of sacred geometry. Despite the evolution in aesthetic, technology and complexity, the underlying concept behind the Opera House and cathedrals of the Middle Ages is the same. Both manmade structures are inspired by sacred geometric forms found in nature.


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87


SACRED GEOMETRY IN ART AND MUSIC

MUSICAL HARMONY We hear sound, and therefore music, by sensing vibrations in the air. Pitch (how high or low a sound is) reflects the vibration’s ‘speed’ or frequency. Stringed instruments allow more than one note to be played at a time, and notes which are played simultaneously on an instrument can sound harmonious or discordant. This is dependent on your musical preferences by in an objective, arithmetic way that underlies vibrating strings and all music. The golden ratio appears in intervals of musical scales as well as in arrangements of musical compositions. What is remarkable is that only whole number ratios produce harmonious results — the ratio of the vibrational frequency of the musical octave is 2:1. If the ratios of the strings were altered to 4.2 or 3.7 units in length, for example, the result would be dissonant. This confirmed Pythagoras’ belief that whole numbers have sacred value. Whole numbers form scales, which are the building blocks of composing and playing music.

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  Sacred Geometry in Modern Culture


.61 8

1

1

.61 8

1

Using the golden section in instrument construction minimizes acoustic resonance and increaes sound quality. As a result, it was used in the original design of violins in order to achieve its supberb tonal qualities.

T H E C H A R AC TER O F MUSIC A L SC A L ES differs within every musical character. A minor scale, which uses the minor third rather than the major third, is often regarded as melancholic. The change, also known as cadence, from a major chord to its minor version is particular haunting. The reverse, from minor to major, known as Picardy third, is regarded as uplifting. Musicians use the character of different scales and cadences to invoke emotion from music, originally discovered by using Pythagoras’ arithmetic. Pentagonic (five-note) scales — occur in folk music worlwide and appear in English hymns and popular music. Adding a note between the fourth and fifth of a minor pentatonic scale — the famous ‘blue note’ in blues, jazz and rock music — also has a very powerful character. A variety of other scales exist, but the principle of whole number ratios always holds true.

89


M ODER N ART & D E S I G N Because of its beautiful nature, mathematics has been a part of art and design for ages and is now an integral parts of a variety of modern art forms, including painting, photography, graphic design and web design. During the Renaissance, the same principles applied to buildings were also applied to paintings. The most famous example was Leonardo da Vinci’s Last Supper, which used the geometry of perspective and also incorporated symbolic numbers. Perspective is also in the modern painting shown above at a sacred geometry exhibit in Lake Tahoe. In terms of photography, the divine proportion can produce aesthetically pleasing compositions that can be magnets for the human sub-conscious. When you take the sweet spot of the Fibonnaci Ratio and recreate it four times into a grid, you get what looks to be a rule of thirds grid. However, upon closer inspection, you will see that this grid is not an exact splitting of the frame into three pieces. Instead of a 3 piece grid that goes 1+1+1=frame, you get a grid that goes 1+.618+1=frame.

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  Sacred Geometry in Modern Culture


The Fibonacci sequence and Corbosier’s Modular are both important sacred ratios in the world of graphic design in terms of page layout. The divine proportion has been used by companies like Apple to design products, it’s said to have been used by Twitter to create their new profile page, and has been used by major companies all over the world to design logos. A designer using a Corbosier’s Modular scale based on the Fibonacci series to determine format and grid may continue with this approach when selecting a range of type sizes. If the increments are considered to be too great, a second proportional scale. Because proportional scales are based on a common unit selected by the designer, such as millimeters, inches or points, the page has an internal harmony in much the same way as the combinations of notes within a single musical chord. Utilizing divine proportions is also common in web design because it is believed to increase legibility, usability and readability of websites. Utilizing the golden ratio when constructing content and layout blocks gives the site balance and provides a sense of closure and structural harmony to the page. So it is clear, then, that as technology, art and music change, the divine proportions have remained constant and will continue to play a major in the fundamentals of living things and in our society.

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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. — Matila Ghyka


INDEX Chartres Cathedral

26-29, 33

Labyrinth

Corbusier, Le

66-67

Logarithmic spiral

Da Vinci, Leonardo

44, 64

Mandelbrot, Benoit

Double helix Euclid Fibonacci sequence

70-71 9

44-45, 50, 51 57 12-13

Modulor

66-67

Music

88-89

Flower of Life

12-13

Parthenon, The

24-25

Flying buttress

26-27

Pascal's Triangle

Fractal geometry

57-61

Phi

Fruit of Life Golden ratio Golden rectangle Golden section Golden spiral Gothic architecture

15-18, 44, 51-52

Metratron's Cube

32-33

12-13 15, 68 18-19, 68 9, 15, 43, 66-69

Phyllotaxis Pisa, Leonardo Plato Platonic solids

17 9, 24, 39, 52, 64-69, 74 44-45, 52 15, 51 9, 10-11, 12 10-11, 12

18-19

Pythagoras

9, 71, 88-89

26-27, 83

Seed of Life

12-13

Grace Cathedral

30-31

Sydney Opera House

Great Pyramid

22-23

Tree of Life

James, John

26-27

Wren, Christopher

80-81, 84-87 12-13 26-27

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COLOPHON Name

Kristin Riger

ID Number

02594374

Course

Typography 3

Professor

Lian Ng

Semester

Fall 2011

Typefaces

Mercury Display Mercury Text Whitney HTF

Photography

Kristin Riger Kyprianos Elisseou Javier Martinez Avedillo Philipp Klinger Simon Bradford

Illustration

Kristin Riger

Paper

Epson Premium Presentation Paper Matte

Printer

Epson Stylus Pro 3880 Inkjet Printer

Binding

Taurus Bookbindery

Divine Proportions: Mathematical Constants in Art & Nature  

Mathematical and geometric constants found in nature and how they are applied in art, design and architecture.

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