Frei Otto

_FreiOtto’sExtensiveresearchandmodelling

_Multihalle,Mannheim

_OlympiastadionOlympicStadium,Munich _StuttgartHighspeedRailwayStation–Stuttgart,Germany

Networks,citiesandtrees-Occupying and connecting Unit DS10 - Kayleigh Dickson

FreiOttostudiedarchitectureinBerlinbeforehewasdraftedafighterpilotinthefinalyearsofWorldWarII.Itwasnotedthat hewasinternedinaFrenchPOWcampand,withhisaviationengineeringtraining,theurgencyforhousingandthelackof materialitwasherehebegantoexperimentwithtentstosheltertheprisoners.AftertheendofthewarhereturnedtostudyingbrieflyintheUnitedStatesafterwhichhevisitedpeoplesuchasErichMendelsohn,MiesvanderRohe,RichardNeutra, and Frank Lloyd Wright. In1952OttobeganaprivatepracticeinGermany,ofwhichhereceivedhisfirstsignificantattentionathisBundesgartenschau saddle-shapedcable-netmusicpavilioninKassel.Shortlyfollowedafterwhenin1954heearnedadoctoratefortensioned constructions Otto challenged the advances in structural m athematicsandcivilengineeringandisaphenomenalmindintheworldoflightweighttensileandmembranestructures, foundingthefamousInstituteforLightweightStructuresattheUniversityofStuttgartin1964,remainingasheadtillhisretirement as university professor. BarryPattenisregardedtohaveinfluencedFreiOtto’sarchitecturalandstructuraldesigns,hismostfamousdesignisthe Myer Music Bowl, Melbourne in 1959. Otto’s project list includes: _The West German Pavilion,1967 Montreal Expo _ Tuwaiq Palace, Saudi Arabia, 1970 _ The roof for the 1972 Munich Olympic Arena, inspired by Vladimir Shukhov’s architecture. _The Japanese Pavilion at Expo 2000, with a roof structure made entirely of paper in collaboration with Shigeru Ban. Awards: _ Thomas Jefferson Medal in Architecture, 1974 _ Wolf Prize in Architecture, 1996-1997 _ RIBA Royal Gold Medal, 2005 Networks, cities and trees - Occupying and connecting Naturesnaturalsystemsconstructorganicandefficientforms,whichwehavechosentooverlookinplaceof‘unnatural’buildingsforanumberofdecades.Wenowdemandfromarchitecturelighter,energyefficientandadaptablebuildingsthatcould grow or shrink with societies needs with out compromising safety, security or quality of life. FreiOttoarguedthatthehumanspontaneousnetworksofurbanityfollowsimilarpatternstothestructureswithinnaturesuch asleaves,insectcoloniesorsoapbubbles.Focusingonenergyanalysisthenetworksarenotformallyplannedandtherefore depict the evolutionary process to create an efficient minimal energy path. FreiOttoevolvedhisresearchthroughextensivemodelstodefine,researchandtestcomplextensileforms.Asthescaleof hisprojectsincreasedheresearchedhowman-madelandscapemightberedefined,notonlyaestheticallybutinitssustainability.Developinganalternativesystemofgridshellstructuresbuiltontheprinciplesofeleganceandstructurallightness.

Images: _ First Row _ Natures natural efficient structure _ Second Row _ _ Third Row _ The Geometry Of Bending, a 2D study

Buckminster Fuller

_ The Geodesic Dome and The Montreal Biosphère, 1967

_TensegritySphere

The lattice shell structures - geodesic domes, which have been used as parts of military radar stations, civicbuildings,environmentalprotestcampsandexhibitionattractions,followsthesamemethodologyas Bauersfeld’s design. Thedomeconstructionisbasedonthebasicprinciplesofbuildingsimpletensegritystructureswhichis lightweightandstable.ThedesignofthegeodesicdomewasaresultofFuller’sexplorationofnature’s constructing principles. In Fiction the Fuller Dome is referenced by John Brunner in his novel Stand on Zanzibar, in which a geodesicdomewassaidtocovertheentireislandofManhattan,floatationonairduetothehot-airballoon effect of the large air-mass under the dome.

BuckminsterFullerpioneeredtermssuchas“ephemeralization”meaning“moreandmorewithlessand lessuntileventuallyyoucandoeverythingwithnothing”andsynergeticswhichreferencedthestudyof systemsintransformationwithemphasisontotalsystembehavior.Hisstudiesleadtotheexplorationof energyandmaterialityprinciplesinArchitecture.Aswellasthetheorizationthat“naturalanalyticgeometry oftheuniversewasbasedonarraysoftetrahedral”,exploredthroughtheclose-packingofspheresandthe number of compressive or tensile members required to stabilize an object in space.

Ephemeralization & Synergetics

Unit DS10 - Kayleigh Dickson

BuckminsterFuller,anAmericanengineer, theorist,author,designer,inventor,futurist,aswellasthesecondpresidentof MensaInternational.Hepublishednearto30booksanddevelopednumerousinventions;ofwhichisthegeodesicdomeishis mostnoted.Carbonmoleculesknownasfullerenesor‘buckyballs’werenamedduetotheircloseresemblancetoBuckminsterFuller’sgeodesicspheres.Buckminsterinitiallystruggledwithgeometry,andtheinabilitytounderstandtheabstraction that‘achalkdotontheblackboardrepresentedamathematicalpoint,orthatanimperfectlydrawnlinewithanarrowonthe endwasmeanttostretchofftoinfinity’.Hewoulddesignitemsfrommaterialshebroughthomefromthewoods,andeven making his own tools, experimenting with designing a new apparatus for human propulsion of small boats. Fuller’seducationstartedatMiltonAcademyinMassachusetts,andleadtoHarvardUniversity,eventhoughheexpelledfrom Harvardtwiceashesawhimselfasanon-conformingmisfitinthefraternityenvironment.LateronhereceivedaSc.D.from BatesCollegeinLewiston,Maine.InanunconventionalmannerforanarchitectheworkedinCanadaasamechanicina textilemill,thenasalaborerinthemeat-packingindustry,ashipboardradiooperatorduringWW1,asaneditorofapublication,andasacrash-boatcommander.By32,BuckmisterFullerwasbankrupt,jobless,andlivinginlow-incomehousingin Chicago,Illinois.In1922withthedeathofhisdaughterheapparentlywassaidtohaveresponsibleforherdeathwhatcaused himtodrinkfrequentlyandtocontemplatesuicide.Somehowhechosetoturnthisexperiencetoembarkon“anexperiment, tofindwhatasingleindividual[could]contributetochangingtheworldandbenefitingallhumanity.”Hisrecoverywasgreatly linkedtohiscommitmentto“thesearchfortheprinciplesgoverningtheuniverseandhelpadvancetheevolutionofhumanity inaccordancewiththem...findingwaysofdoingmorewithlesstotheendthatallpeopleeverywherecanhavemoreand more.”In 1928, Fuller accepted a job decorating the interior of the café in exchange for meals, lecturing several times a week,andlatermodelsoftheDymaxionhousewereexhibitedatthecafé.InacollaborationwithIsamuNoguchi,Fullersoon designed the Dymaxion car. OneifnotthemostnotableprojectofBuckminsterFullerswasfoundedattheBlackMountainCollegeinNorthCarolina,1948 and1949,wherewiththesupportofagroupofprofessorsandstudents,hebeganreinventingaprojectthegeodesicdome. Theconceptofthegeodesicdomehadbeencreatedaround30yearsearlierbyDr.WaltherBauersfeld,thoughduetoFuller’s popularizing this type of structure he was awarded United States patents. In1949,hebuilthisfirstgeodesicdomebuildingthatcouldsustainitsownweightandmeasuredat4.3metersindiameter andconstructedofaluminumaircrafttubingandavinyl-plasticskin,intheformofanicosahedron.Toprovehisdesign,and toawenon-believers,seemingslightlyeccentricFullersuspendedfromthestructure’sframeworkseveralstudentswhohad helped him build it. Within a few years there were thousands of these domes around the world. Fullerhadapassionforbetteringlifethroughhisdesignsanddevelopedmanyideas,designsandinventions,particularly regardingpractical,inexpensiveshelterandtransportation.Hedocumentedhislife,philosophyandideasscrupulouslyby adailydiarywhichwastobecalledDymaxionChronofile,andconsistedoftwenty-eightpublicationsdocumentinghislife from1915to1983,equatingtoapproximately270feet(82m)ofpapers.Fullerthendesignedoneofthemostefficientrepresentationoflandmassthroughanalternativeprojectionmap,calledtheDymaxionmap.ThiswasdesignedtoshowEarth’s continents with minimum distortion when projected or printed on a flat surface. Philosophy and worldview Fullerstronglybelievedinthinkingglobally,andheexploredprinciplesofenergyandmaterialefficiencyinthefieldsofarchitecture,engineeringanddesign.Andwasdeeplyconcernedwithsustainabilityandabouthumansurvivalundertheexisting socio-economicsystem,yetremainedoptimisticabouthumanity’sfuture.“Selfishness,”toBuckminsterFullerwasdeemed as,“unnecessaryandhence-forthunrationalizable....Warisobsolete.”TohimforUtopiatowork,hethoughtthatautopia needed to include everyone. BuckmisterFullerwasincrediblyquirkyandwasnotedtobeafrequentflier,whowhencrossingtimezonesheworethree watches;oneforthecurrentzone,oneforthezonehehaddeparted,andoneforthezonehewasgoingto.InfactThe“Omega IncablocOysterAccutron72”BuckminsterFullerdesignedthecase.Whilstflyinghealsoinsertedasheetofnewsprintover a shirt and under a suit jacket, to provide completely effective heat insulation during long flights. Heexperimentedwithpolyphasicsleep,whichhecalledDymaxionsleep.In1943,hetoldTimeMagazinethathehadslept onlytwohoursadayfortwoyears.Hequittheschedulebecauseitconflictedwithhisbusinessassociates’sleephabits,but stated that Dymaxion sleep could help the United States win World War II.

_Tensegrity Dome

Experimentation_01 Differential Growth, Edge Effects and Other Interactions

Material

Rule Theory

Unit DS10 - Kayleigh Dickson

_Cracked Mud Cracked Surface Rule

Throughthestudyoftwo-dimensionalnetworksinthehexagonalcell,inthestudyofbiologicalsystemsitisimportanttostudy growthsequences,sequentialsolidificationofformsand“edgeeffects”asthiswillgreatlyeffecttheoutcomeofthefinalform.

3-connectedverticeswillappearwithsomeareasappearingtoformat90 degrees–withlatercracksthenappearingperpendiculartotheoriginal crack.Thoughtheconfigurationmaybeformedwiththreelinesmeetingat a central point at 120 degrees, an isotropic force or growth.The isotropic force or isotropy will produce a simple net and a form of a three-rayedhexagonal pattern.

Crackingorshrinkageoccursinaccordancewithstrictrules.Thiswillnormallyoccurwithanoriginalfaultlineinthematerial. Thetensilestresswillactevenlyinalldirectionsisgreateronthetop.,whichwillacceleratethelineargrowthofthe‘cracks’. SideCrackswilltakeformatrightangles,startingfromaweakpointinthewallofamaterial.Ifthiscrackinthesideofthewall wastoconnectwithanexistingcrack,thentheyareorientatedbythetensilestressinparalleltothewallsurfaceat90degrees.Ifthematerialwastocontractinuniformthenahexagonalshapewillappear,andthreelineswillmeetat120degrees 1_withcrackedsurfacessuchasceramicglaze,mud,etc.,the‘cracking’effectwillnotformimmediatelyyetitwillappear over time 2 _ the surface of a heated aluminum sheet will show the formation at early stages of melting, is unlike the two-dimensional formation showing with soap bubbles.

_ Heated Aluminum Sheet Heated Aluminum Sheet Rules

4 _ the grain boundaries in Niobium forms through the appearance of etching and shows a lack of uniformity.

Theformationnotonlyfollowstheruleofthreelinesmeetingatonesingular pointbutyouwillcommonlyfindthatthepointwillbesurroundedbyangle of 120 degrees.

_ Soap Bubbles Soap Bubbles Rules The four vertices are meet at approximately 109 degrees. They must be in juxtaposition as to fill all the space and their interfaces must conform to the laws of surface tension.

_Grain Boudaries in Niobium Grain Boundaries in Niobium Rules Here you would see few exceptions to the three-rayed vertex rule in which the point is surrounded by an angle of 120 degrees. Each grain is a polyhedral crystal which will mesh close to another crystal in a closest packed array. The intrinsic forces will confront the extrinsic forces exerted from another neighbouring crystal

Material Net Formation

Detail

Visual Rule

Experimentation_02 Foam Fun! TheNaturalFormation of Bubbles and The Effects of Freezing Conditions Unit DS10 - Kayleigh Dickson

Sculpted Foam Fun! Thesoapbubbleholdsitsformasasurfacelayerofaliquidhasacertainsurfacetension,whichcausesthelayertoactlike anelasticsheet.Thefilmofthesoapbubbleisextremelyflexibleandcanproducewavesbasedontheforceexerted.HoweverasIfoundoutifabubblemadesolelyofpureliquidaloneisnotstableandsoapisneededtostabilizeabubble.WhenI madethesoapfoamby‘hand’thefoamcontainedlevelsofsoapandwaterthatweretooheavyforthestructureofthefoam tosupportanditwouldcollapseunderitsownweight.Thesecondattemptattheexperimentinvolvedhigh-pressurewaterto createthefoam,whicheffectivelybuiltalighterstructureandwasabletosupportweight,cantilevering,etc.tobesculptedinto aform.Thefoamhasanadhesivequalitythatenabledittotakeitsform,thoughespeciallywiththeheaviersolutionasthe foambrokeitsbondsinareasvoids,spacesandenvironmentsappeared.Fromtheanalysisgainedthroughthisexperiment Iplantopushthetheoryfurther,whichtheintentiontofocusonthelighterfoamandthesespaceswhicharenaturallyformed through the materials strengths and weaknesses.

Natural Foam Fun! ForthesecondexperimentationattemptImixedthefoamsolutioninabathwhichproducedalighterfoamandthereforea greaterstrengthfoam,asItransferredthefoamInoticedthatitnaturallydemonstratedanorganicstructureofitsownchoosing. The photo documentation notes this unintentional form. TheThirdExperimentwhichiscurrentlyon-goinginvolvesplacingthefoamsolutionintoafreezertodocumenttheeffectof theadverseconditionsontheform.Ifsoapbubblesarekeptinanenvironmentwithatemperature−15°C(5°F)orbelow itwillfreeze,thoughiftheyweretobebroughtintoawarmerenvironmentitwillcausethebubbletocrumbleunderitsown weight at a rapid weight. Photo documentation of this experiment is to follow.

Experimentation_2D The Geometry of Bending

2D_Bending from a singluar point

Bending Lines Slenderortwo-dimensionalobjectschangetheirformundertheinfluenceofappliedforceactingtransversetotheiraxis.The resultofbucklingmayoccurtothestructureduetoalackofstabilityinrelationtotheaxialforces.Thecharacteristicofbendinglinescanbemathematicalformulatedaslongasthemateriallawsweretobeabidedby.Thedeformationconfiguration canbederivedfromthetypeofload,withthestiffnessandlengthinfluencinggreatlytheoutcomeofthebendingmovement. Parabola Ifarodorelasticmaterialwhichhasaconstantcrosssectionandalinearforce-elongationcurveisloadedwithasingleforce then the bending line is equal to a third-order parabola.

2D Line Experiments by Frei Otto

Unit DS10 - Kayleigh Dickson

Circular Arc Circularbendinglinesdevelopwhenfixed-endmomentsactinginoppositedirectionsareactiveattheendsofanelasticrod. If the two ends are to be held to one another then the bending resistance remains constant and a circle is formed. Combined Forms

Physical Analysis of Grasshopper

GrasshopperExperiments

Typicalbendinglinescanbeproducedbycombiningdifferenttypesofload,inarchitecturethesemayresultinnumerous characteristicbuildingforms.Sincenaturallygrownrodsareusedforbuildingwhichareseldomandprismatic,thebuilding will adapt to the bending line of conical rods.

2D_Bending from two points

2D_Bending from three points

2D_Bending from two points

2D_Bending from one point with a fixed centre

DigitalisedGeometry Delaunay Triangulation & Voronoi Diagram Unit DS10 - Kayleigh Dickson

Delaunay triangulation TheDelaunayTriangulationwasinventedbyBorisDelaunayin1934.TheDelaunayTriangulationisusedinmathematics andcomputationalgeometryforasetPofpointsintheplaneisthetriangulationDT(P)suchthatnopointinPisinsidethe circumcircleofanytriangleinDT(P).ThepointoftheDelaunaytriangulationsmaximizetheminimumangleofalltheangles of the triangles in the triangulation; they tend to avoid skinny triangles. ForasetofpointsonthesamelinethereisnoDelaunaytriangulation(thenotionoftriangulationisdegenerateforthiscase). Forfourormorepointsonthesamecircle(e.g.,theverticesofarectangle)theDelaunaytriangulationisnotunique:eachof thetwopossibletriangulationsthatsplitthequadrangleintotwotrianglessatisfiesthe“Delaunaycondition”,i.e.,therequirement that the circumcircles of all triangles have empty interiors.

The DTFE consists of three main steps: Step 1 Thestartingpointisagivendiscretepointdistribution.Intheupperleft-handframeofthefigureapointdistributionisplottedinwhichatthecenteroftheframeanobjectislocatedwhosedensity diminishesradiallyoutwards.InthefirststepoftheDTFEtheDelaunaytessellationofthepointdistributionisconstructed.Thisisavolume-coveringdivisionofspaceintotriangles(tetrahedrain threedimensions),whoseverticesareformedbythepointdistribution(seefigure,upperright-handframe).TheDelaunaytessellationisdefinedsuchthatinsidetheinteriorofthecircumcircleofeach Delaunay triangle no other points from the defining point distribution are present.

Byconsideringcircumscribedspheres,thenotionofDelaunaytriangulationextendstothreeandhigherdimensions.GeneralizationsarepossibletometricsotherthanEuclidean.HoweverinthesecasesaDelaunaytriangulationisnotguaranteed to exist or be unique.

d-dimensional Delaunay ForasetPofpointsinthe(d-dimensional)Euclideanspace,aDelaunaytriangulationisatriangulationDT(P)suchthatno pointinPisinsidethecircum-hypersphereofanysimplexinDT(P).Itisknown[2]thatthereexistsauniqueDelaunaytriangulation for P, if P is a set of points in general position; that is, there exists no k-flat containing k + 2 points nor a k-sphere containing k + 3 points, for 1 ≤ k ≤ d − 1 (e.g., for a set of points in ℝ3; no three points are on a line, no four on a plane, no four are on a circle, and no five on a sphere). TheproblemoffindingtheDelaunaytriangulationofasetofpointsind-dimensionalEuclideanspacecanbeconvertedtothe problemoffindingtheconvexhullofasetofpointsin(d+1)-dimensionalspace,bygivingeachpointpanextracoordinate equalto|p|2,takingthebottomsideoftheconvexhull,andmappingbacktod-dimensionalspacebydeletingthelastcoordinate.Astheconvexhullisunique,soisthetriangulation,assumingallfacetsoftheconvexhullaresimplices.Nonsimplicial facetsonlyoccurwhend+2oftheoriginalpointslayonthesamed-hypersphere,i.e.,thepointsarenotingeneralposition.

Step 2 TheDelaunaytessellationformstheheartoftheDTFE.Inthefigureitisclearlyvisiblethatthetessellationautomaticallyadaptstoboththelocaldensityandgeometryofthepointdistribution:where thedensityishigh,thetrianglesaresmallandviceversa.Thesizeofthetrianglesisthereforeameasureofthelocaldensityofthepointdistribution.ThispropertyoftheDelaunaytessellationisexploitedinstep2oftheDTFE,inwhichthelocaldensityisestimatedatthelocationsofthesamplingpoints.Forthispurposethedensityisdefinedatthelocationofeachsamplingpointastheinverse of the area of its surrounding Delaunay triangles (times a normalization constant, see figure, lower right-hand frame). Step 3 In step 3 these density estimates are interpolated to any other point, by assuming that inside each Delaunay triangle the density field varies linearly

_DelaunayTriangulation

Ref: http://en.wikipedia.org/wiki/Delaunay_triangulation

Relationship with the Voronoi Diagram TheDelaunaytriangulationofpointsetPingeneralpositioncorrespondstothedualgraphoftheVoronoitessellationforP. Special cases include the existence of three points on a line and four points on circle. The Delaunay triangulation with all the circumcircles and their centers is een to the left. Connecting the centers of the circumcircles produces the Voronoi diagram is seen to the right.

Voronoi diagram In the simplest case, we are given a set of points S in the plane, which are theVoronoi sites. Each site s has aVoronoi cell, alsocalledaDirichletcell,V(s)consistingofallpointsclosertosthantoanyothersite.ThesegmentsoftheVoronoidiagram areallthepointsintheplanethatareequidistanttothetwonearestsites.TheVoronoinodesarethepointsequidistantto three (or more) sites.

AgraphicalrepresentationoftheVoronoidiagramfromtwo-dimensiontothree-dimension

Ingeneral,thesetofallpointsclosertoapointcofSthantoanyotherpointofSistheinteriorofa(insomecasesunbounded) convexpolytopecalledtheDirichletdomainorVoronoicellforc.ForeachpointxinStakethesetofpointsclosertocthan tox(asdescribedabove).ThentaketheintersectionofallthesesetstogettheVonoroicellforc.Thesetofsuchpolytopes tessellatesthewholespace,andistheVoronoitessellationcorrespondingtothesetS.Ifthedimensionofthespaceisonly 2,thenitiseasytodrawpicturesofVoronoitessellations,andinthatcasetheyaresometimescalledVoronoidiagrams. Ref: http://en.wikipedia.org/wiki/Voronoi_diagram

_ Voronoi Diagram

MarcFornesdesignwasinspiredbyVoronoiDiagramtheory

Approximate Voronoi diagram of a set of points.Theblendedcolorsrepresentedthe blurred boundary of the Voronoi cells.

Experimentation_3D The Geometry of Bending

Bending from two points in 2D

Theory

Multiple Influenced Bending

Influenced Bending Form

Natural Bend Form

Rig

Unit DS10 - Kayleigh Dickson Rig System

Process in 3D

Experimentational Experimentation_3D Process The Geometry of Bending

Unit DS10 - Kayleigh Dickson

Experimentation_3D The Geometry of Bending

Unit DS10 - Kayleigh Dickson

Experimentation_3D The Geometry of Bending

Unit DS10 - Kayleigh Dickson

Digital Confirmation_01 The Geometry of Bending

Unit DS10 - Kayleigh Dickson

Digital Confirmation_02 The Geometry of Bending

Unit DS10 - Kayleigh Dickson

Singular Anchor Point

Two Point Movement

Singular Anchor Point

_ Comparing physical 2D Bening compared to the Grasshopper model

Generative Rules Generative path systems based on nature’s angles Unit DS10 - Kayleigh Dickson

The rig aids in the manipulation of ply strips, through holding fixed the two points of bending control on the strip allowing for an angle of degree to be permanent, unless soaked in boiling water when it will return to its original form. 90 Degrees

Generative Paths In generative path systems, users, when close to paths connecting at a right angle of the second generation tend to take a short cut as soon at the ‘T-junction’ becomes visible. This is especially true when the way not only to the next occupier but also the occupier beyond is sought.

The theory of generative paths has similarities to my studies into the geometry of bending, where the curve in the bending of ply is not tight to point 0,0, but actually rounds on to the new angle prematurely of point 0,0. So when analyzing the strength in nature’s angles I can approach the experimentation with this theory and rigorously test multiple angles. To analyse if nature does produce the strongest geometry of bending or if does not is it the most efficient structure?

33 Degrees

Would a structure become stronger if in this experiment the material was set at a required angle, with multiple strips then placed up against each other tightly, before pouring hot water over the top, creating compression on the neighbouring strip?

Natural Rule Analysis_01

Unit DS10 - Kayleigh Dickson

Patterns and Grids The advantage of the hexagonal grid resulting from the triangular grid has a broad spectrum of application. Any expansion and subdivision is possible. Natural Patterns When layers of mud shrink due to drying, the non-cracked expanses can be considered as >>territories<<. The majority of these >>territories<< are hexagonal. The key points of the surfaces form a clear triangular grid. Often, many generations can be seen, with an increasing number of pentagonal surfaces.

Natural and Technological Occupations Occupations belonging in the areas of non-living nature, living nature and technology can be distinguished. Geodesy or the division of the Earthâ€™s surface can be considered a planned, i.e. less natural occupation mechanism. The division of the Earthâ€™s surface in to meridians and circles of latitude, for instance, is both artificial and at the same time useful.

_ Cracked Mud

Almost all natural occupations are subject to self-constituting principles of varying strength. This is especially clear in the <<occupation>> of an even surface by shrinkage cracks (in clay or glazes), which predominately enclose hexagonal surfaces whose key points, in an ideal situation, form a triangular pattern. _ The cracked mud out line that was imput in to grashopper for testing and analysis

_ Grasshopper Definition

_ The Voronoi output in grasshopper of the cracked mud which tests if the 120 degree rule applies

_ Theout put from grasshopper which shows that the cracked mud very closely applies natures rules of deformation

Circle Packing Analysis_01

Unit DS10 - Kayleigh Dickson

Circle packing The most efficient way to pack different-sized circles together is not obvious. In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that all circles touch another. The associated “packing density”, η of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres. The branch of mathematics generally known as “circle packing”, however, is not overly concerned with dense packing of equal-sized circles (the densest packing is known) but with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like. While the circle has a relatively low maximum packing density, it does not have the lowest possible. The “worst” shape to pack onto a plane is not known, but the smoothed octagon has a packing density of about 0.902414, which is the lowest maximum packing density known of any centrally-symmetric convex shape.[1] Packing densities of concave shapes such as star polygons can be arbitrarily small.

Packings in the plane The centers of three circles in contact form an equilateral triangle, making this a hexagonal packing.

Identical circles in a hexagonal packing arrangement, the densest packing possible. In two dimensional Euclidean space, Joseph Louis Lagrange proved in 1773 that the lattice arrangement of circles with the highest density is the hexagonal packing arrangement,[2] in which the centres of the circles are arranged in a hexagonal lattice (staggered rows, like a honeycomb), and each circle is surrounded by 6 other circles. The density of this arrangement is Axel Thue provided the first proof that this was optimal in 1890, showing that the hexagonal lattice is the densest of all possible circle packings, both regular and irregular. However, his proof was considered by some to be incomplete. The first rigorous proof is attributed to László Fejes Tóth in 1940.[2] At the other extreme, very low density arrangements of rigidly packed circles have been identified.

Packings on the sphere

Packings in bounded areas

Unequal circles

A related problem is to determine the lowestenergy arrangement of identically interacting points that are constrained to lie within a given surface. The Thomson problem deals with the lowest energy distribution of identical electric charges on the surface of a sphere. The Tammes problem is a generalisation of this, dealing with maximising the minimum distance between circles on sphere. This is analogous to distributing non-point charges on a sphere.

Fifteen equal circles packed within the smallest possible square. Only four equilateral triangles are formed by adjacent circles.

A compact binary circle packing with the most similarly sized circles possible.

Packing circles in simple bounded shapes is a common type of problem in recreational mathematics. The influence of the container walls is important, and hexagonal packing is generally not optimal for small numbers of circles.

There are also a range of problems which permit the sizes of the circles to be non-uniform. One such extension is to find the maximum possible density of a system with two specific sizes of circle (a binary system). Only nine particular radius ratios permit compact packing, which is when every pair of circles in contact is in mutual contact with two other circles (when line segments are drawn from contacting circlecenter to circle-center, they triangulate the surface).

Circle Packing Analysis_02

Unit DS10 - Kayleigh Dickson

Kepler conjecture

Kepler conjecture

Face-centered cubic packing

Face-centered cubic packing

The Kepler conjecture, named after the 17th-century German astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is slightly greater than 74%.

The Kepler conjecture, named after the 17th-century German astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is slightly greater than 74%.

In 1998 Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Hales’ proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees have said that they are “99% certain” of the correctness of Hales’ proof, so the Kepler conjecture is now very close to being accepted as a theorem.

In 1998 Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Hales’ proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees have said that they are “99% certain” of the correctness of Hales’ proof, so the Kepler conjecture is now very close to being accepted as a theorem.

Imagine filling a large container with small equal-sized spheres. The density of the arrangement is the proportion of the volume of the container that is taken up by the spheres. In order to maximize the number of spheres in the container, you need to find an arrangement with the highest possible density, so that the spheres are packed together as closely as possible.

Imagine filling a large container with small equal-sized spheres. The density of the arrangement is the proportion of the volume of the container that is taken up by the spheres. In order to maximize the number of spheres in the container, you need to find an arrangement with the highest possible density, so that the spheres are packed together as closely as possible.

Experiment shows that dropping the spheres in randomly will achieve a density of around 65%. However, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a hexagonal lattice, then put the next layer of spheres in the lowest points you can find above the first layer, and so on – this is just the way you see oranges stacked in a shop. At each step there are two choices of where to put the next layer, so this natural method of stacking the spheres creates an uncountably infinite number of equally dense packings, the best known of which are called cubic close packing and hexagonal close packing. Each of these arrangements has an average density of

Experiment shows that dropping the spheres in randomly will achieve a density of around 65%. However, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a hexagonal lattice, then put the next layer of spheres in the lowest points you can find above the first layer, and so on – this is just the way you see oranges stacked in a shop. At each step there are two choices of where to put the next layer, so this natural method of stacking the spheres creates an uncountably infinite number of equally dense packings, the best known of which are called cubic close packing and hexagonal close packing. Each of these arrangements has an average density of

The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density.

The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density.

The conjecture was first stated by Johannes Kepler (1611) in his paper ‘On the six-cornered snowflake’. He had started to study arrangements of spheres as a result of his correspondence with the English mathematician and astronomer Thomas Harriot in 1606. Harriot was a friend and assistant of Sir Walter Raleigh, who had set Harriot the problem of determining how best to stack cannon balls on the decks of his ships. Harriot published a study of various stacking patterns in 1591, and went on to develop an early version of atomic theory.

The conjecture was first stated by Johannes Kepler (1611) in his paper ‘On the six-cornered snowflake’. He had started to study arrangements of spheres as a result of his correspondence with the English mathematician and astronomer Thomas Harriot in 1606. Harriot was a friend and assistant of Sir Walter Raleigh, who had set Harriot the problem of determining how best to stack cannon balls on the decks of his ships. Harriot published a study of various stacking patterns in 1591, and went on to develop an early version of atomic theory.

Nineteenth century Kepler did not have a proof of the conjecture, and the next step was taken by Carl Friedrich Gauss (1831), who proved that the Kepler conjecture is true if the spheres have to be arranged in a regular lattice. This meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one. But eliminating all possible irregular arrangements is very difficult, and this is what made the Kepler conjecture so hard to prove. In fact, there are irregular arrangements that are denser than the cubic close packing arrangement over a small enough volume, but any attempt to extend these arrangements to fill a larger volume always reduces their density. After Gauss, no further progress was made towards proving the Kepler conjecture in the nineteenth century. In 1900 David Hilbert included it in his list of twenty three unsolved problems of mathematics—it forms part of Hilbert’s eighteenth problem.

Six circles theorem Some examples of theorem configuration changing the radius of the first circle. In the last configuration the circles are pairwise coincident. In geometry, the six circles theorem relates to a chain of six circles together with a triangle, such that each circle is tangent to two sides of the triangle and also to the preceding circle in the chain. The chain closes, in the sense that the sixth circle is always tangent to the first circle. The name may also refer to Miquel’s six circles theorem, the result that if five circles have four triple points of intersection then the remaining four points of intersection lie on a sixth circle.

Nineteenth century Kepler did not have a proof of the conjecture, and the next step was taken by Carl Friedrich Gauss (1831), who proved that the Kepler conjecture is true if the spheres have to be arranged in a regular lattice. This meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one. But eliminating all possible irregular arrangements is very difficult, and this is what made the Kepler conjecture so hard to prove. In fact, there are irregular arrangements that are denser than the cubic close packing arrangement over a small enough volume, but any attempt to extend these arrangements to fill a larger volume always reduces their density. After Gauss, no further progress was made towards proving the Kepler conjecture in the nineteenth century. In 1900 David Hilbert included it in his list of twenty three unsolved problems of mathematics—it forms part of Hilbert’s eighteenth problem.

Circle Packing Analysis_02

Unit DS10 - Kayleigh Dickson

Malfatti circles Apollonian gasket

In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle. Malfatti’s problem has been used to refer both to the problem of constructing the Malfatti circles and to the problem of finding three area-maximizing circles within a triangle.

In mathematics, an Apollonian gasket or Apollonian net is a fractal generated from triples of circles, where each circle is tangent to the other two. It is named after Greek mathematician Apollonius of Perga. An Apollonian gasket can be constructed as follows. Start with three circles C1, C2 and C3, each one of which is tangent to the other two (in the general construction, these three circles can be any size, as long as they have common tangents). Apollonius discovered that there are two other non-intersecting circles, C4 and C5, which have the property that they are tangent to all three of the original circles – these are called Apollonian circles (see Descartes’ theorem). Adding the two Apollonian circles to the original three, we now have five circles.

Malfatti’s problem Comparison of the Malfatti circles and the three area-maximizing circles within an equilateral triangle In 1803 Gian Francesco Malfatti posed the problem of cutting three cylindrical columns out of a rectangular wedge of marble, maximizing the total volume of the columns. He assumed, as did many others after him, that the solution to this problem was given by three tangent circles within the triangular cross-section of the wedge. That is, more abstractly, he conjectured that the three Malfatti circles have the maximum total area of any three disjoint circles within a given triangle.

Malfatti circles

In a similar way we can construct another new circle C7 that is tangent to C4, C2 and C3, and another circle C8 from C1, C3 and C1. This gives us 3 new circles. We can construct another three new circles from C5, giving six new circles altogether. Together with the circles C1 to C5, this gives a total of 11 circles.

Malfatti published in Italian and his work may not have been read by many in the original. It was popularised for a wider readership in French by Joseph Diaz Gergonne in the first volume of his ``Annales” (1810/11), with further discussion in the second and tenth. However, this advertisement most likely acted as a filter, as Gergonne only stated the circle-tangency problem, not the area-maximizing one. The conjecture is wrong; Lob and Richmond (1930), who went back to the original Italian text, observed that for some triangles a larger area can be achieved by a greedy algorithm that inscribes a single circle of maximal radius within the triangle, inscribes a second circle within the largest of the three remaining corners of the triangle, and inscribes a third circle within the largest of the five remaining pieces. The difference in area for an equilateral triangle is small, just over 1%, but as Howard Eves pointed out in 1946, for an isosceles triangle with a very sharp apex, the optimal circles (stacked one atop each other above the base of the triangle) have nearly twice the area of the Malfatti circles.

Continuing the construction stage by stage in this way, we can add 2·3n new circles at stage n, giving a total of 3n+1 + 2 circles after n stages. In the limit, this set of circles is an Apollonian gasket. The Apollonian gasket has a Hausdorff dimension of about 1.3057. Curvature The curvature of a circle (bend) is defined to be the inverse of its radius.

First Ajima–Malfatti point

Goldberg (1967) showed that, for every triangle, the Lob–Richmond procedure produces three circles with larger area than the Malfatti circles, so the Malfatti circles are never optimal. Zalgaller and Los’ (1994) classified all of the different ways that a set of maximal circles can be packed within a triangle; using their classification, they proved that the greedy algorithm always finds three area-maximizing circles, and they provided a formula for determining which packing is optimal for a given triangle. In his 1997 Ph.D. thesis, Melissen conjectured more generally that, for any integer n, the greedy algorithm finds the area-maximizing set of n circles within a given triangle; the conjecture is known to be true for n ≤ 3.

Apollonian sphere packing An Apollonian gasket can also be constructed by replacing one of the generating circles by a straight line, which can be regarded as a circle passing through the point at infinity. Alternatively, two of the generating circles may be replaced by parallel straight lines, which can be regarded as being tangent to one another at infinity. In this construction, the circles that are tangent to one of the two straight lines form a family of Ford circles.

The regular hexagonal packing is the densest sphere packing in the plane. (1890) The 2-dimensional analog of the Kepler conjecture; the proof is elementary. Henk and Ziegler attribute this result to Lagrange, in 1773 (see references, pag. 770).

The three-dimensional equivalent of the Apollonian gasket is the Apollonian sphere packing. Symmetries

The hexagonal honeycomb conjecture

The dodecahedron conjecture

▪ Negative curvature indicates that all other circles are internally tangent to that circle. This is bounding circle ▪ Zero curvature gives a line (circle with infinite radius). ▪ Positive curvature indicates that all other circles are externally tangent to that circle. This circle is in the interior of circle with negative curvature. Variations

Thue’s theorem

The most efficient partition of the plane into equal areas is the regular hexagonal tiling. Hales’ proof (1999).

Take one of the two Apollonian circles – say C4. It is tangent to C1 and C2, so the triplet of circles C4, C1 and C2 has its own two Apollonian circles. We already know one of these – it is C3 – but the other is a new circle C6.

Comparison of the Malfatti circles and the three area-maximizing circles within an equilateral triangle

If two of the original generating circles have the same radius and the third circle has a radius that is twothirds of this, then the Apollonian gasket has two lines of reflective symmetry; one line is the line joining the centres of the equal circles; the other is their mutual tangent, which passes through the centre of the third circle. These lines are perpendicular to one another, so the Apollonian gasket also has rotational symmetry of degree 2; the symmetry group of this gasket is D2.

Bubble Occupation Study Bubble Formation

Unit DS10 - Kayleigh Dickson

Bubbles and mountain passes Scientists had thought bubbles form when jostling liquid molecules create pockets of low density in the liquid containing relatively fewer molecules than surrounding regions. Most of the time, other molecules will just rush in to fill in these air pockets. However, an exodus of molecules can also occur, causing the pockets, or bubbles, to grow. David Corti, a chemical engineer at Purdue University in Indiana, compares the process to scaling a mountain. A pocket of air begins at the bottom of one side of the mountain (the liquid phase) and must climb the mountain and reach a destination on the other side (the vapor phase) to become a bubble. “A small bubble needs to climb up one side of the mountain, cross through a reasonably well-defined mountain pass before it rolls down the other side of the mountain towards forming very large bubbles,” Corti explained. According to the conventional view, once the bubble makes it over the pass, it tumbles down the other side of the mountain like a snowball, picking up more molecules and growing bigger.

Surface Tension and Bubbles

The new computer simulation suggests there “is no other side of the mountain,” Corti told LiveScience. “Once it gets over the pass, we have found that the mountain just disappears, in a sense.”

The surface tension of water provides the necessary wall tension for the formation of bubbles with water. The tendency to minimize that wall tension pulls the bubbles into spherical shapes (LaPlace’s law).

Rather than having to descend another slope, the bubble just plummets directly into the vapor phase. As a result, bubble formation might occur more rapidly than previously thought, Corti said. Broader pathways

The pressure difference between the inside and outside of a bubble depends upon the surface tension and the radius of the bubble. The relationship can be obtained by visualizing the bubble as two hemispheres and noting that the internal pressure which tends to push the hemispheres apart is counteracted by the surface tension acting around the circumference of the circle.

The new research also suggests the metaphorical mountain pass is actually more broad and flat than previously thought, allowing for several possible pathways from the liquid to the vapor phase. “In the traditional view … there are only a few pathways that go through the pass,” Corti said. “From our work, we have shown that it is in fact quite broad, so that there are a large number of pathways that will lead over the mountain top.”

For a bubble with two surfaces providing tension, the pressure relationship is:

The findings could have implications for how scientists predict the rate of bubble formation, and could help improve safety for industries that rely on bubbles, the researchers say. “We are still working out the full implications of this ourselves,” Corti said Bubble Pressure Experiments with Bubbles When many small bubbles are randomly distributed on a surface of water they will join together to form an expanse, until a singular raft if formed.

The net upward force on the top hemisphere of the bubble is just the pressure difference times the area of the equatorial circle:

If the small bubbles are the same size, they take on a structure, which is hexagonal when seen from above. The minimal territories are therefore formally identical with those of distancing occupation. The territory centres arrange themselves spontaneously into a grid of equilateral triangles.

The surface tension force downward around circle is twice the surface tension times the circumference, since two surfaces contribute to the force:

Even if the small bubbles, i.e. the narrowest occupations are of unequal size the pattern remains similar. Hexagonal >>meshes<< in triangular (non-equilateral) structures predominately. Pattern formation in attractive occupation is therefore identical to that of distancing occupation, with only one difference: with attractive occupation, the greatest density is reached.

This gives

This latter case also applies to the case of a bubble surrounded by a liquid, such as the case of the alveoli of the lungs.

Bending Analysis_01

Unit DS10 - Kayleigh Dickson

Bending Analysis_02

Unit DS10 - Kayleigh Dickson

Experimentation_3D The Geometry of Bending Pushing, Flexing and Testing Unit DS10 - Kayleigh Dickson

Bubble Form Versus Voronoi Bubble Formation 3 Unit DS10 - Kayleigh Dickson

Bubbles and mountain passes Scientists had thought bubbles form when jostling liquid molecules create pockets of low density in the liquid containing relatively fewer molecules than surrounding regions. Most of the time, other molecules will just rush in to fill in these air pockets. However, an exodus of molecules can also occur, causing the pockets, or bubbles, to grow. David Corti, a chemical engineer at Purdue University in Indiana, compares the process to scaling a mountain. A pocket of air begins at the bottom of one side of the mountain (the liquid phase) and must climb the mountain and reach a destination on the other side (the vapor phase) to become a bubble. “A small bubble needs to climb up one side of the mountain, cross through a reasonably well-defined mountain pass before it rolls down the other side of the mountain towards forming very large bubbles,” Corti explained.

Cracking Experiment _ 01 The extracted form piped in Grasshopper

According to the conventional view, once the bubble makes it over the pass, it tumbles down the other side of the mountain like a snowball, picking up more molecules and growing bigger.

The grasshopper definition which formed the Voronoi bubble formation below.

2

The new computer simulation suggests there “is no other side of the mountain,” Corti told LiveScience. “Once it gets over the pass, we have found that the mountain just disappears, in a sense.” Rather than having to descend another slope, the bubble just plummets directly into the vapor phase. As a result, bubble formation might occur more rapidly than previously thought, Corti said. Broader pathways The new research also suggests the metaphorical mountain pass is actually more broad and flat than previously thought, allowing for several possible pathways from the liquid to the vapor phase. “In the traditional view … there are only a few pathways that go through the pass,” Corti said. “From our work, we have shown that it is in fact quite broad, so that there are a large number of pathways that will lead over the mountain top.” The findings could have implications for how scientists predict the rate of bubble formation, and could help improve safety for industries that rely on bubbles, the researchers say.

Cracking Experiment _ 02

“We are still working out the full implications of this ourselves,” Corti said

1

Experiments with Bubbles

Extracted 2D polylines

When many small bubbles are randomly distributed on a surface of water they will join together to form an expanse, until a singular raft if formed. If the small bubbles are the same size, they take on a structure, which is hexagonal when seen from above. The minimal territories are therefore formally identical with those of distancing occupation. The territory centres arrange themselves spontaneously into a grid of equilateral triangles.

3 _ Voronoi output from grasshopper based on the central points of each closed polygon of the cracked mud

Even if the small bubbles, i.e. the narrowest occupations are of unequal size the pattern remains similar. Hexagonal >>meshes<< in triangular (non-equilateral) structures predominately. Pattern formation in attractive occupation is therefore identical to that of distancing occupation, with only one difference: with attractive occupation, the greatest density is reached.

2 _ Closed Polygon Outline of the experiment 1 _ Cracking Experiment _ 01 Form Cracking Experiment _ 03

The progress from the block voronoi form built in grasshopper, into the extracted face edges which leaves a simpler form, before being piped in to a 3D form

Burnt Mud Experimentation_01

Unit DS10 - Kayleigh Dickson

Mudcracks (also known as desiccation cracks or mud cracks) are sedimentary structures formed as muddy sediment dries and contracts. Naturally forming mudcracks start as wet, muddy sediment desiccates, causing contraction via a decrease in tensile strength. Individual cracks join up forming a polygonal, interconnected network. These cracks may later be filled with sediment and form casts on the base of the overlying bed. Syneresis cracks are broadly similar features that form from subaqueous shrinkage of muddy sediment caused by differences in salinity or chemical conditions, rather than subaerial exposure and desiccation. Syneresis cracks can be distinguished from mudcracks because they tend to be discontinuous, sinuous, and trilete or spindle-shaped Cracking Experiment _ 02

Cracking Experiment _ 03

Cracking Experiment _ 04

Regular

Crack Filling Simple

Complete Complete

Compound

Orientated

Irregular

Incomplete Incomplete

Random

Crack Shape

Unbridged

Complete

Incomplete

Bridged

Non-Orthogonal

Cracking Experiment _ 01

Orthogonal

Naturally forming mudcracks start as wet, muddy sediment desiccates, causing contraction via a decrease in tensile strength[citation needed]. Individual cracks join up forming a polygonal, interconnected network. These cracks may later be filled with sediment and form casts on the base of the overlying bed.[citation needed] Syneresis cracks are broadly similar features that form from subaqueous shrinkage of muddy sediment caused by differences in salinity or chemical conditions,[1] rather than subaerial exposure and desiccation. Syneresis cracks can be distinguished from mudcracks because they tend to be discontinuous, sinuous, and trilete or spindle-shaped

Burnt Mud Experimentation_02 Non-Orthogonal

Orthogonal

Orthogonal

Crack Filling

Unit DS10 - Kayleigh Dickson

Environments and substrates Naturally occurring mud cracks form in sediment that was once saturated with water. Abandoned river channels, floodplain muds, and dried ponds are localities that form mudcracks.[5] Mud cracks can also be indicative of a predominately sunny or shady environment of formation. Rapid drying, which occurs in sunny environments, results in widely spaced, irregular mud cracks, while closer spaced more regular mud cracks indicates a shady formation environment.[6] Polygonal crack networks similar to mudcracks can form in man-made materials such as ceramic glazes, paint film, and poorly made concrete. Similar features also occur in frozen ground, lava flows (as columnar basalt), and igneous dykes and sills.

The term fold is used in geology when one or a stack of originally flat and planar surfaces, such as sedimentary strata, are bent or curved as a result of permanent deformation. Synsedimentary folds are those due to slumping of sedimentary material before it is lithified. Folds in rocks vary in size from microscopic crinkles to mountain-sized folds. They occur singly as isolated folds and in extensive fold trains of different sizes, on a variety of scales. Folds form under varied conditions of stress, hydrostatic pressure, pore pressure, and temperature - hydrothermal gradient, as evidenced by their presence in soft sediments, the full spectrum of metamorphic rocks, and even

Plan View as primary flow structures in some igneous rocks. A set of folds distributed on a regional scale constitutes a fold belt, a common feature of orogenic zones. Folds are commonly formed by shortening of existing layers, but may also be formed as a result of displacement on a non-planar fault (fault bend fold), at the tip of a propagating fault (fault propagation fold), by differential compaction or due to the effects of a high-level igneous intrusion e.g. above a laccolith.

Compound

Section View

Plan View Section View

Plan View Section View

Mud cracks can be preserved as v-shaped cracks on the top of a bed of muddy sediment or as casts on the base of the overlying bed. When they are preserved on the top of a bed, the cracks look as they were at the time of formation. When they are preserved on the bottom of the bedrock the cracks are filled in with younger, overlying sediment. In most bottom-of-bed examples the cracks are the part that sticks out most. Bottomof-bed preservation occurs when mud cracks that have already formed and are completely dried are covered with fresh wet sediment and buried. Through burial and pressure the new wet sediment is further pushed into the cracks where it dries and hardens. The mud cracked rock is then later exposed to erosional factors.[2] In these cases, the original mud cracks will erode faster than the newer material filling in the spaces. These types of mud cracks are useful for geologists to determine the vertical orientation of rock samples that have been altered through folding or faulting.

Simple

Preservation

The axial surface is the surface defined by connecting all the hinge lines of stacked folding surfaces. If the axial surface is a planar surface then it is called the axial plane and can be described by the strike and dip of the plane. An axial trace is the line of intersection of the axial surface with any other surface. Finally, folds can have, but donâ€™t necessarily have a fold axis. A fold axis, â€œis the closest approximation to a straight line that when moved parallel to itself, generates the form of the fold.â€? . A fold that can be generated by a fold axis is called a cylindrical fold. This term has been broadened to include near-cylindrical folds. Often, the fold axis is the same as the hinge line

Burning Man

Unit DS10 - Kayleigh Dickson

What is Burning Man?

WHAT IS BURNING MAN?

What is Burning Man? As have so many others who’ve struggled to answer that difficult question, this section offers a comprehensive overview of the event and its history.

Trying to explain what Burning Man is to someone who has never been to the event is a bit like trying to explain what a particular color looks like to someone who is blind. In this section you will find the peripheral definitions of what the event is as a whole, but to truly understand this event, one must participate. This site serves to try to paint a picture of the Burning Man experience to those who are new to the project, as well as to give those participants looking to keep the fire burning in their daily lives an environment in which to connect to their fellow community members. For a brief yet eloquent overview of the entire event from the time of arrival to the time of exodus, please read “What is Burning Man?”, an essay written by participant and one-time web team member, Molly Steenson. Please see archived sections for each year to read more about the art themes, art installations and theme camps for each year.

Ten Principles or The 10 core guiding concepts of the Burning Man project: Radical Inclusion Anyone may be a part of Burning Man. We welcome and respect the stranger. No prerequisites exist for participation in our community. Gifting Burning Man is devoted to acts of gift giving. The value of a gift is unconditional. Gifting does not contemplate a return or an exchange for something of equal value. Decommodification In order to preserve the spirit of gifting, our community seeks to create social environments that are unmediated by commercial sponsorships, transactions, or advertising. We stand ready to protect our culture from such exploitation. We resist the substitution of consumption for participatory experience. Radical Self-reliance Burning Man encourages the individual to discover, exercise and rely on his or her inner resources. Radical Self-expression Radical self-expression arises from the unique gifts of the individual. No one other than the individual or a collaborating group can determine its content. It is offered as a gift to others. In this spirit, the giver should respect the rights and liberties of the recipient. Communal Effort Our community values creative cooperation and collaboration. We strive to produce, promote and protect social networks, public spaces, works of art, and methods of communication that support such interaction. Civic Responsibility We value civil society. Community members who organize events should assume responsibility for public welfare and endeavor to communicate civic responsibilities to participants. They must also assume responsibility for conducting events in accordance with local, state and federal laws. Leaving No Trace Our community respects the environment. We are committed to leaving no physical trace of our activities wherever we gather. We clean up after ourselves and endeavor, whenever possible, to leave such places in a better state than when we found them. Participation Our community is committed to a radically participatory ethic. We believe that transformative change, whether in the individual or in society, can occur only through the medium of deeply personal participation. We achieve being through doing. Everyone is invited to work. Everyone is invited to play. We make the world real through actions that open the heart. Immediacy Immediate experience is, in many ways, the most important touchstone of value in our culture. We seek to overcome barriers that stand between us and a recognition of our inner selves, the reality of those around us, participation in society, and contact with a natural world exceeding human powers. No idea can substitute for this experience

Here you will find links that will take you on a trip through the past - through the history of Burning Man - from its early days on a small beach in San Francisco through its evolution into the bustling city of some 48,000+ people that the Burning Man event has become today. These people make the journey to the Black Rock Desert for one week out of the year to be part of an experimental community, which challenges its members to express themselves and rely on themselves to a degree that is not normally encountered in one’s day-to-day life. The result of this experiment is Black Rock City, home to the Burning Man event. There are no rules about how one must behave or express oneself at this event (save the rules that serve to protect the health, safety, and experience of the community at large); rather, it is up to each participant to decide how they will contribute and what they will give to this community. The event takes place on an ancient lakebed, known as the playa. By the time the event is completed and the volunteers leave, sometimes nearly a month after the event has ended, there will be no trace of the city that was, for a short time, the most populous town in the entire county. Art is an unavoidable part of this experience, and in fact, is such a part of the experience that Larry Harvey, founder of the Burning Man project, gives a theme to each year, to encourage a common bond to help tie each individual’s contribution together in a meaningful way. Participants are encouraged to find a way to help make the theme come alive, whether it is through a large-scale art installation, a theme camp, gifts brought to be given to other individuals, costumes, or any other medium that one comes up with. The Burning Man project has grown from a small group of people gathering spontaneously to a community of over 48,000 people. It is impossible to truly understand the event as it is now without understanding how it has evolved. See the first years page and Burning Man 1986 - 1996 for the legendary story of Burning Man’s beginnings and to understand how the event has come to become what it is today. The timeline gives a short overview of what each year looked like. Please also check out the detailed archives for years 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, and 2010. Within each of these years are descriptions each year’s art theme, theme camps, large art installations, as well as maps, journals of our city being built, the newsletters to the community for each year, issues of the Black Rock Gazette (a daily news publication produced and printed on the playa), and clean up reports for each year, including a list of those sites that failed to “leave no trace”. These pages help understand the larger scope of the entire experience, from the planning that happens year-round to make each event possible, to the clean-up efforts which take place for sometimes months after the city has disappeared. The impact of the Burning Man experience has been so profound that a culture has formed around it. This culture pushes the limits of Burning Man and has led to people banding together nation-wide, and putting on their own events, in attempt to rekindle that magic feeling that only being part of this community can provide. The Black Rock Arts Foundation promotes interactive art by supporting public art that exists outside the event, and has a special interest in supporting art at regional events. Additionally, Burning Man has over two thousand volunteers who work before, during and after the event (many who work year-round) to make the event a reality. To give of your time and talents, please see the Participate section of the website.

Burning Man

Unit DS10 - Kayleigh Dickson

_ Burning Man Pavilion 1

_ Burning Man Pavilion 2

_ Burning Man Pavilion 3

WHAT IS BURNING MAN? Trying to explain what Burning Man is to someone who has never been to the event is a bit like trying to explain what a particular color looks like to someone who is blind. In this section you will find the peripheral definitions of what the event is as a whole, but to truly understand this event, one must participate. This site serves to try to paint a picture of the Burning Man experience to those who are new to the project, as well as to give those participants looking to keep the fire burning in their daily lives an environment in which to connect to their fellow community members. For a brief yet eloquent overview of the entire event from the time of arrival to the time of exodus, please read “What is Burning Man?”, an essay written by participant and one-time web team member, Molly Steenson. Please see archived sections for each year to read more about the art themes, art installations and theme camps for each year. Here you will find links that will take you on a trip through the past - through the history of Burning Man - from its early days on a small beach in San Francisco through its evolution into the bustling city of some 48,000+ people that the Burning Man event has become today. These people make the journey to the Black Rock Desert for one week out of the year to be part of an experimental community, which challenges its members to express themselves and rely on themselves to a degree that is not normally encountered in one’s day-to-day life. The result of this experiment is Black Rock City, home to the Burning Man event. There are no rules about how one must behave or express oneself at this event (save the rules that serve to protect the health, safety, and experience of the community at large); rather, it is up to each participant to decide how they will contribute and what they will give to this community. The event takes place on an ancient lakebed, known as the playa. By the time the event is completed and the volunteers leave, sometimes nearly a month after the event has ended, there will be no trace of the city that was, for a short time, the most populous town in the entire county. Art is an unavoidable part of this experience, and in fact, is such a part of the experience that Larry Harvey, founder of the Burning Man project, gives a theme to each year, to encourage a common bond to help tie each individual’s contribution together in a meaningful way. Participants are encouraged to find a way to help make the theme come alive, whether it is through a large-scale art installation, a theme camp, gifts brought to be given to other individuals, costumes, or any other medium that one comes up with. The Burning Man project has grown from a small group of people gathering spontaneously to a community of over 48,000 people. It is impossible to truly understand the event as it is now without understanding how it has evolved. See the first years page and Burning Man 1986 - 1996 for the legendary story of Burning Man’s beginnings and to understand how the event has come to become what it is today. The timeline gives a short overview of what each year looked like. Please also check out the detailed archives for years 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, and 2010. Within each of these years are descriptions each year’s art theme, theme camps, large art installations, as well as maps, journals of our city being built, the newsletters to the community for each year, issues of the Black Rock Gazette (a daily news publication produced and printed on the playa), and clean up reports for each year, including a list of those sites that failed to “leave no trace”. These pages help understand the larger scope of the entire experience, from the planning that happens year-round to make each event possible, to the clean-up efforts which take place for sometimes months after the city has disappeared. The impact of the Burning Man experience has been so profound that a culture has formed around it. This culture pushes the limits of Burning Man and has led to people banding together nation-wide, and putting on their own events, in attempt to rekindle that magic feeling that only being part of this community can provide. The Black Rock Arts Foundation promotes interactive art by supporting public art that exists outside the event, and has a special interest in supporting art at regional events. Additionally, Burning Man has over two thousand volunteers who work before, during and after the event (many who work year-round) to make the event a reality. To give of your time and talents, please see the Participate section of the website.

_ Burning Man Pavilion 4 - Preburn

_ Burning Man Pavilion 4 - BURN

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