Deployable Structures Inspired by the Origami Art FEB 2 7 2004 LIBRARIES

DeployableStructures Inspired by the OrigamiArt by KenGiesecke

BachelorofArts, University of Pennsylvania, 1998

Submittedtothe Department ofArchitecture in partial fulfillment ofthe requirements forthe degree of Masterof Architecture atthe Massachusetts Institute ofTechnology

February2004 @2004Ken Giesecke. Allrights reserved.

Theauthor hereby grants to MITpermission to reproduce and to distribute publiclypaper and electroric copes of thisthesis document in whole or in part.

signatureof author:_ KenGiesecke DepartmentofArchitecture January16. 2004 certifiedby:_ JohnOchsendorf AssistantProfessor, DepartmentofArchitecture ThesisSupervisor

CarolBurns VisitingAssociate Professor, Department ofArchitecture ThesisSupervisor acceptedby: BillHubbard, Jr. MASSACHUSEITSINST11~UTE AdjunctAssociate Professor ofArchitecture Chairman,Department OF TECHNOLOGY CommitteeonGraduate Students

FEB2 7 2004 ROTCH

LIBRARIES thesis committee

ThesisSupervisors

JohnOchsendorf AssistantProfessor, Department ofArchitecture

2 CarolBurns VisitingAssociate Professor, Department ofArchitecture

ThesisReaders

ShunKanda SeniorLecturer, Dept. of Architecture

MartinDemaine VisitingScientist, Lab for Computer Science DeployableStructures Inspired by the Origami Art byKen Giesecke

Submittedto theDepartment ofArchitecture in partial fulfillment of therequirements for thedegree of Masterof Architectureat the Massachusetts Institute of Technology February2004

Mythesis is anexploration of designmethods and tools using origami as a vehicleto testtheir usefulness abstract andcoming to termswith their limitations. I have taken my fascination with a particulardevelopment in origamiand put my belief in itspotential for architectural application to thetest by way of variousinvestiga- tions:materials and structural analysis, mathematical reasoning, manipulating space and form, parametric modeling,fabrication, and finite element testing. Partingfrom conventional, figural forms, mathematicians developed open-surface forms together with theo- remsthat governed the ability of thesefolded forms to foldflat. I selecteda particularform, the Kao-fold, forits simplicity, beauty, and structural properties and imagined many exciting possibilities, specifically for its applicationindesigning a deployable structure. I analyzedits crease pattern, exploring variations and their corresponding folded forms. Simultaneously,different material ideas for larger-scalestructures were tested and a particularconfiguration wasassessed for internalstresses and its structural stability. Its transformation from a flatsheet to a foldedstate was scrutinized under the lens of mathematicalreasoning, namely trigonometry, bylinking the acuteangle of itscrease pattern and the dihedral angle in its foldedstate to itsfinal folded configuration. Therigidity of thisinvestigation was offset by the freedom afforded in manipulatingpaper models. As such,different spatial qualities and forms were explored while addressing the issue of scaleand potential applications.The transformational characteristics discovered were digitally simulated via theconstruction of parametricmodels, which was a morecontrolled manipulation of the form in a virtualspace. In orderto go beyondthe realm of representationandaddress real-life building issues, a temporaryopen-air shelter was designedand constructed indetail. The goal was to tacklethe complexity of assigningmaterials, designing componentsand fabricated them. As a finalendeavor, the model's construction was tested for its structural stabilityusing a finiteelement software.

ThesisSupervisors: John Ochsendorf Title: Assistant Professor, Department ofArchitecture CarolBurns Title: VisitingAssociate Professor, Department ofArchitecture tomy advisor, John Ochsendorf, forthe consistent enthusiasm hebrought toevery discussion. It has helped me tremendously to stay guided and motivated

acknowledgementtoCarol, my co-advisor, foraddressing my assumptions and my shortcom- ings

tomy readers, Shun Kanda and Martin Demaine

tomy unofficial reader, Erik Demaine. Your interview inthe New Scientist ledme to pursueapplications offolding in a seminarwhich pavedthe wayto my thesis

4 toprofessor Kenneth Kao who gave the seminar and for teaching us to thinkcritically

toAlex for generously giving me so much of histime and energy toward realizingthe "sphere model"

to Susanand Victor for showing me the ropes to Catia

toother members of the "quiet studio"- Mike, Karim, Chris, Aliki and Tracy fortheir moral support

to Mrs.Marvin E Goody for support of myresearch and work. 03 abstract

06 motivation tableof contents forresearch 08 precedentstudies

10 origami:investigating open surface forms

14 addressingmaterials: a preliminary structural analysis

20 a mathematicalinvestigation: comprehendingthetransformation bytrigonometry

papermodels: a physical manipulation of space & form

parametricmodeling: digital manipulation through defining parameters

fabrication:beyond diagrams &representations

finiteelement analysis: an assessment ofstructural stability

conclusion

comments&criticisms

bibliography

imagecredits motivation for research

I developedmy interest in deployablestruc- turesand origami in a seminartaken in the semesterpreceding my thesis at theGradu- ateSchool of Designin Harvardtaught by professorKenneth Kao titled GSD 6410 Con- ceptto Construction: Building Components &Assemblies. The course examined and dis- cussed variouscasestudies of instancesin architecturewhere truly innovative thinking in theengineering ofa structuralcomponent enabledunique design solutions toemerge. Thework of engineers/designers suchas 6 JeanProuvet, BuckminsterFuller,and Peter Ricewere examined for their inventiveness. Inthis context,students were taught to thinkcritically about current-day design and buildingtechnologies and try to bringnew insightinto their use as a starting pointfor innovation.Eventually, this led to thefinal assignmentforwhich I researched collaps- iblefurniture and structures and in parallel exploredbasic foldingpatterns inorigami. Theresearch was guidedbythe belief in origamito beable to inform the design of deployablestructures and focused on the designpossibilities a particularof pleated form(which I eventually named the Kao-fold asno inventor could be identified).

precedent Ageodesic dome uses a pattern studies of self-bracingtriangles in a patternthat gives maximum structuraladvantage, thus theo- The ideasand thoughts sur- reticallyusing the least material possible.(A 'geodesic"line on a roundingsome of the projects sphereis the shortestdistance discussedinthe course as well betweenany two points.) asthose researched formy finalassignment have carried 8 overinto the development of mythesis. Certain precedent studiesthat have been more inspiringmay be presented:

In RenzoPiano's IBM pavilion,a series of polycarbonatepyramids arereinforced with cast aluminumconnections (6. splicedand glued into laminatedbeechwood.

(6.) CommissionedbyRice Univer- sityArt Gallery, Bamboo Roof is anoutdoor site-specific installa- tioncreated by Japanese Archi- tectShigeru Ban. The workfeatures an expansive , 4 open-weavecanopy of bamboo 4 boardsthat spans the Gallery Plaza.

Metallic frarin Metalicangle

Blow oint

jointdetails for an accordion roof

Water-tight hingesdrive thedesign in a motor- drivenaluminum sheet roof.

(7.) (8.) BiaxialFlasher Supreme by origami:investigating UshioIkegami. Thethicker lines represent open-surfaceforms mountainfolds.

Startinginthe 1930's, the art of origami beganto extend beyond traditional, figural workswith which we most strongly associate withorigami. Liapi, 386 Enthusiasts' devel- / - opedan interest in the mathematical proper- tiesfound within the crease patterns.The creasepattern is literallythe pattern of creasesseen in the paper oncetheorigami formis unfoldedinto a flatsheet. (seeexamples shown to the right) (8a.) CubicFlahser by UshioIkegami. Extremelydificult to fold, the cubic Thus,working backwards, one is ableto flasherdoes not expand and con- 10 determinethe final form for a particular (9 tract smoothilylikea basicflasher. designonce the creasepattern is known. ( Theonly key required to unlockingthis DNA whichdetermines a certain form is to know whethereach individual crease within the patternresults in a mountainora valleyfold. Ifeither side of the crease is folded towards you,that is a valleyfold and if away,you get a mountainfold. (see image on bottom-right)

Inthe creasepattem shown, divid- inga squareinto four equal parts, allthe creases representvalley folds. Onecategory oforigami, which developed outof thisnewfound interest in origami's mathematicalproperties ismodular or unit origami.Unit origami, as it'sname describes, isformed from a numberof repeating units The6-unit wedge is whichlink to formvarious closed shapes, formedby a seriesof modularbelt-units. whichin turn, may be linked to oneanother withcertain foldedunits. (see image)

Asecond development was open-surface forms.These open-surface forms had cap- turedmy interest, as they seemed to carry great potentialforbeing applied as architec- turalelements, be it anouter skin of a build- (10.) ing,an interior surface anorentire habitable structure.Together with the development of theoremswhich governed the ability of these open-surfaceforms to befolded flat, one may imagineexciting possibilities fororigami's a.At any vertex, the sum of the anglesis360 degrees applicationindesigning deployable struc- b.The number of creases originating ata vertex must always be even. tures- structuresthat could be collapsed, c.(the Hushimitheorem) When four creases meet a at common point, the transported,and efficiently deployed. Thesetheorems were derivedbymathe- differencebetween two adjacentangles has to be equal to the difference maticiansthrough a careful understanding betweenthe remaining two angles. forhow the crease pattern of thepleated d.(the Kawasaki theorem) The angles al, a2,a3 .... , a2n,surrounding a single formcould be manipulated togive specific vertexin a flat origami crease pattern must satisfy the following requirement: resultingforms. They determine the underly- ingmathematical properties offolded forms al + a3 + a5 + ... + a2n-1= 180degrees whichallowed them to fold flat. Liapi, 387 and Thesetopological conditions that any pattern a2+ a4+ a6+ ... + a2n= 180degrees of creasesmust fulfill are: Onesuch open-surface form Thered and blue lines represent mountainfolds and the green, thatwas of particular interest valleyfolds, Variations inthe wasthe Mars fold, patented Marsfold are basedon two pos- byTaborda Barreto. The Mars siblepositions for the rhombuses. Whileall patterns fold flat, theonly foldis composed ofsquares restrictionisthat the lines forming andrhombuses, lying adjacent eachobtuse angle of the rhombs tothe squares. What was so befolded in thesame direction. fascinatingabout the Mars fold washow its simple crease pat- terncould yield such a variety offorms depending onthe nil directionofthe folds.

Yet,the problem with these formswas that they often requiredlarge amounts of 12 bendingand twisting both to developthe form and to keep it inits folded state. If the struc- turewere to be constructed on a largerscale, a materialother thanpaper would not be as for- giving. Whilethe Mars fold and other open-surfaceforms could be shapedwithout any distortion withinthe surface it was also vitalthat one could give the pleatedsurface a degree of curvature. Thissecond characteristic (whichcan not be found in thefishbone or tessellatedstars fold)was important inbeing ableto containspace and offer a senseof enclosure. This cur- vaturecould be implemented intothe pineapple fold, however,it wouldnot translate into anefficient structure as there is a significantamount of overlap- pingbetween the panels. Ultimately,I returned to the Kao-foldfor its versatility, beauty,but most importantly, itsinherent structural value. Amongother open-surface 13 forms,it hasa clearlyidentifi- ableload pathand a consid- erabledepth to its section. It remindsone of a truss,whereby thegreater the depth of the section,the stronger it becomes andthe greater the distance it mayspan. However, as theform is not composed of linearelements, differing from thetruss, the panels offer cov- erageand allow one to imagine a varietyof uses. addressingmaterials: a preliminarystructural analysis

DEVELOPED Afterplaying with different configurations SHAPE ofthe Kao fold in paper,I was eager towork with materials besides paper that wouldbegin to suggest the difficulties of workingon a largerscale. Different material/ assemblyideas were tried with the goal of FOLDEDSHAPE keepingthe paper model's ability to expand (11.) andcollapse freely. In looking at different materialpossibilities, it soon became appar- 14 entthat paper could not be substituted with a singlesheet material but that it wouldbe necessaryto make a compositewhere the materialalong the crease lines of the paper modelwould be flexible differing from the restof the form that could be rigid. For this flexiblematerial along the fold-lines, drywall andvinyl tape were tested. In the panels, chipboard,acrylic, and thin plywood were tried.The elastic vinyl tape was used for, Thechip model formed by two 90degree modulesjoined at their tofold the structure,thematerial in order endswas part of a studyby C.G. wouldneed to bestretched atthe nodes. Fosterwith S. Krishnakumar in Ascan be seen in thechipboard and dry- 1986of a family of foldable,por- tablestructures based on triangu- was walltape model, another solution to larpanels. simplyextract material at thenodes thereby disconnectingoneseam from the next. (11b.) 15 I assignedthe Kao fold the form ofa half-archas a startingpoint tounderstand how thestresses fromits own dead weight are carriedthrough the structure to theground. The makingof a half-archchip model using zip- lockbags (polyethylene) along theseams was particularly instructivein being able to see howthe stresses caused by the materialsown dead weight were carriedthrough from one panel tothe next, that is, along the seams. Aheavier plywood model was 16 madeto increasethe stresses onthe seams and thereby clar- ifyexactly how theseams were resistingthe dead load of the panels.Furthermore, given that thepanels are much larger than thechip model, the connections between themandthe seams couldbe elaborated. Grooves wererouted out so that an inter- stitialstrip of woodcould be sandwichedbetween two layers andhold the sheet material alongthe entire edge. Thetransfer of weight from the topof the arch tothe base couldbe abstracted using short steelrods and rope. From this abstraction,it becomes clearer as tohow the internal stresses insidethe panels are operating, thatis, in compression along thefold and in tension on the othertwo sides. (see sketch) The steelmodels, moreover, areinstructive in identifying the Kao-fold'sweaknesses asa structure.Inthe shape of a half- arch,a tremendousmoment forceexists at the base.Fur- thermore,one can observe that, 17 laterally,it is very unstable(as it wouldsway quite freely with a slightforce to theside!) Afirst reaction toincrease lateralstability would be to completethe arch, thereby transmittingthegravitational forcesfrom one end to the other.The model was con- structedusing two half-arches connectedbya pinjoint at the ends......

Onemeans to stabilize the structure in thelateral direction with external sup- portswould be to haveopposing ten- sionforces applied to it, in theway a youngtree is harnessed tothe ground.

Inobserving the internal forces of anotherabstract model of a slender steelrod held in tensionby a string, anotherconsideration wasto use embeddedpre-tensioned cables which wouldforce the structure toerect itself whilecounteracting thatforce with externaltension cables running diago- nallyfor lateral stability, from the top of thearch tothe ground. 18 Inaccounting forthe moment force at thebase, however, it would be a more stablestructure if the embedded ten- sioncables were brought on the outside,again running diagonally, and, for sakeof elegance, the counteracting cablesbe placed along the inside surface. 19 A Mathematical Investigation: comprehending the transformation bytrigonometry

Simultaneoustothinking about 11..5 acuteangle materialsand structural stability, I wantedto come to terms withthe mathematical principles 20 behindthe transformationofthe Kaofold. That is, to determine therelationship between the geometryof the crease pattern andits corresponding folded form. Afterplaying with paper models it becameapparent that the size ofthe acute angle in thecrease patterncontrolled the degree of curvatureinthe pleated arch. Thesmaller the acute angle,the "tighter"the curl of archwould Furthermore,inlooking at thesection, it maybe deduced that in order for a half- -4-0" archto standperpendicular to the ground plane(In this state, the bottom segment ofthe archis perpendiculartothe ground plane and thetop segment parallel to it.Also the vertices ofthe half-arch are tangent to acircle), the acuteangles would need to addup to 90 degrees. Asthe image shows, the smaller the acute angleis, thegreater are the number of folds.As a completedarch, structurally, the Kao-foldbehaves much like a trussin a half-circleformation. (see image 12) 5.625 Justas "increasingthe modularincre- mentationof thetruss decreases the abil- ityof thevertices to resist concentrated 21 loads",the more faceted the surface of theKao-fold, it loses depth compromising itsstrength. Pearce, p.27 Thus based on anintuitive assessment of the stability of thestructure in accordancewith its depth, it wasdetermined that the 11.25-degree modelwould be developed.

*Thesize of thearched structure is dependentonthe length of theseams. Eacharch shown in theimage was sizedtohouse 2-3 people.

(12.) It isimportant tonote that the study onlyassesses the impact of the acuteangles on the contained space seenin section. That is, there is noconsideration forthe impactof a--V-. thedihedral angle. As the dihedral anglewould affect thecurvature of thearch as well, its mathematical \y relationshiphad to be determined. Inother words, the perpendicular B2 B A conditionachieved previously was based the on assumption that the cos45* = xly formwas folded flat and, as such, tan11.250 = x/I r T 7 cos 450 = 7.161"/y should thearch be deployed in a tan11.250 = x/36" y = 10.13" x = 7.161" semi-unfoldedstate, theeffect of thedihedral angle would need to tan0= yll 22 beaccounted for. By expanding and tan0 = 10.13/36" collapsinga paper model, we learn = 15.7* thatas the arch folds flat, it acquires a tightercurvature and vice-versa. Intuitively,one may presume that to Thefollowing wayto compensateforthis affect, in order isa calculatethe acute angle, given that all units in thefolded piece are in thesame folded state i.e. 60 to maintaina semi-unfolded perpen- degreesor 90 degrees, so that the bottom segment ofthe archis perpendicularto the ground dicularstate, the acute angle of the planeand the topsegment parallel to it. hasbeen creasepattern must be increased It previouslydeduced that the acute angle must be a factorof 90( 22.5,11.25, etc.)to allow for this condition. from11.25 degrees to somegreater However,these calculations depend on the arch being in a fully folded/closedstate. number.Through some basic trig- Theless folded the arch, the onometrywe find that, should the greaterits tendency tobe "less arch like" andbecome a straight, flat form. otherwords, the arch dihedralangle be set at 90 degrees, In "rises",losing its curvature. Therefore,order to compensate thisnew value for the acute angle is in forthis effect, the acute angle would need to beslightly greater thansay...22.5 Through some 15.7degrees. (see page 20) basictrigonometry theproblem may be solved allowing for the designof arches with various curvatures while controlling the degree of folding......

L------C ------R

Forexample, lets say, in the above folded pair of triangles, the acute angle RLB is 11.25degrees when the triangles are completely folded andthe length of the folded edge is determined. In its closed state, we draw a linefrom point B to themidpoint of LRor C which gives usa righttriangle LCB. Nextwe find the length of BC.(In its closed state, we can draw a linethrough B and C which is perpendicular to LR.) Then we open the fold by a determinedangle, say 60 degrees. The diagram below shows the fold in section.

B2 B A 23

\ / \ I

Usingthe value of BC,we can calculate B2C the "new" length of BC once the fold is opened 60 degrees. Then we work backwards and calculate theacute angle given our "new" length of BCor B2C. Theillustration onp.20 shows the two corresponding creasepatterns with the acuteangles of 11.25and 15.7 degrees aswell as their completelyfolded state. Whencompletely foldedwecan see the impactof theacute angle. That is, the higherthe value, thetighter the curvature,and thus, if wewere to unfold and expandthe arch by setting the dihedral angleat 90, the arch opens up and risesto the aforementioned perpendicu- larcondition. This condition was mod- eledby inserting square pieces of chip into thecenter of every folded unit dem- onstratingthe logic and clarity of mathematical thinking. 24 Tomake things more interesting, I col- lapsedone end of the half-arch configuration.The result is a formthat resemblesanorange peel as the first curvaturealong the lengthofthe arch issupplemented bya secondcurvature alongthe arch's width. In this model, essentially,what is happeningis that thedihedral angles arediminishing from oneend of thearch to the other. Using the samethought process and trigono- metriccalculations asbefore, one may determinethe value for each dihedral anglein a semi-unfoldedperpendicular state. The 11.25-degree,perpendicular condition dictates that in a halfarch, there will be three full folded panels and two, half folded panels. Inthe image for the crease pattern, the arch extends an additional 25%, giving four full folded panels. Once the smallest dihedral angle isdetermined, thedifference between its corresponding acute angle and the next pair is established. (Ifone wants to work with the 11.25-degree,perpendicular condition, the smallest acute angle cannot be less than 11.25 degrees, as the planes con not close in on themselves!)Inthis design, I chose 1.636 degrees. Once all the acute angles are determined, their corresponding dihedral angle is calculatedusing the same, right-angle principles asbefore but going backwards. That is whyit isimportant that the acuteangles are paired, thatis, to be able to use the same method as before.

59.358'

newY valuebased on 4.688* decreasing lengthof changinglength of 93.356* dihedralangle dihedralangle $

1 tan11.260* =y 36' ->y= 7167" - cos$= 7.161'17.167"-->0= 2.344* tan 11.25"=x/36 -- x 7.161" 8.) 11.26" - 2.344* x2 - 4.688* 7 1 92 7.) 12.896* - 29.679* x2 - 59.358* tan 12.896*y/35. 81' ->y 8 " ->cos 7.117"/8.192"-> =29.679* 29.679*x2 -59.358" tan 11.25*=x/35.781" -> x = 7.117" 103.454' 6.)12.896" - 5.) 14.532" - 39.88* x2 --- 79.76" tan 14.532*= y135.531"-> y 9.21' -> cos4 = 7.067"/9.21'-> $ = 39.88* 4.) 14.532" - 39.88* x2 -79.76* tan 11.25*= x/35.531' -> x = 7.067" 3.) 16.168*-46.678* x2- 93.356* 11 39" 2.) 16.168"- 46.678* x2 - 93.356* tan16.168* = y/35.258" -> y = 10.222"-> cos p= 7.013"/10.222"->V =46.678* 1.) 17.804"- 51.727* x2 - 103.454 tan 11.25'=x035.258" -- x = 7.013" -1.)17.804" -51.727* x2 -103.454 tan 17.804*= y/34.945"-> y= 11.222"-> cos , = 6.951"/11.222"-> q = 51.727* x2 - 111.39" -2.)19.44* -55.695 tan 11.25"=x/34.945" -> x = 6.951" 1 I acute angle dihedralangle T tan19.440" =y/34.617" ->y= 12.218"-- cos$ =6.886"/12.218'->) = 55.695" tan 11.25"=x/34.617' -> x = 6.886" Whilethis investigation allowed meto establish the relationship betweenthe 2-dimensional creasepattern and its corre- spondingfolded form through mathematicalreasoning, the precisionrequired was limiting andfar removed from the freedomofworking inpaper models.

I hadattempted toapply the math to themaking of a 3d,digital model. Unfortunately, while in theory, the mathematical reasoningissound, due to rounding-offthe values within the calculations, wesee a tightfit betweenpanels at the base andas we work our waytowards the top of the arch, the marginal errors become accumulate and are progressively transparent. papermodels: a physicalmanipulation of space& form

inorder to explore the habitational qualities,addressing scale and 27 potentialapplications, I returned to thefreedom found in working with papermodels. Inexploring forms, I lookedfirst for design ideas using thebasic crease pattern of the Kao- fold.The first gesture to suggest occupationwould be to simplyinsert a scalefigure. And so I tookthe half-archand orange-peel configura- tionsand modeled them as such. Anotheridea would be to mergethe qualitiesofthe full-arch configuration andthe orange-peel and collapse thefull arch at the ends and link the verticeswith linear members. Subsequently,to go further, I identifiedspecific characteris- 2.000 - 0~2~ ticsof the2-dimensional crease pattern,namely that of the axis- OV of thedihedral angle and length 014 of unit.(By axis, what is referred 0'3

to arethe lines along which every V-73 0'4k folded unitlies, collinear with the verticesof the dihedral angles.) Thefollowing diagrams show variationsof thesetwo charac- teristicsand, for some, their cor- z000, - respondingfolded forms.The 2.000* useof 1/8"scale figures imme- diatelygave the sense for thehabitational/spatial quality of 28 theseforms. (seeopposite page)

III I 2000* - II I asor- / , 3.000*- 2.500-~'

3 50*- 4s00o--- 0-4 I I 2

14 T2" 0.4 0'1. \II II

Al ______29 Toaddress a certain scale of constructionand different appli- cationsfor theKao-fold, col- lageswere made as well. The half-archconstruction could be usedfor booths in a carnival orcraft fair, the full arch could bea monumentalentry-way. If theKao-fold was constructed outof steel, it couldserve a morepermanent function muchlike Calder'sout-door sculptures,orif cast-in-place concrete,we may think of FelixCandela's entryway tothe PabellonMusic School, or the 30 interiorof his church in Migrosa (14) (14a)

(13) parametricmodeling: digitalmanipulation through defining parameters

Parallelto my exploration of space and form by using the hands to manipulatepaper models,a different quality of freedom was found in thedigital manipulation ofform. Whileplaying in paper, a certaininsight on the transformational qualities was gained, thatis, to understand exactly how the individual folded units move in relationto each I Parallelplanes offset from each other were otheras they contract and expand. As such,must be stressed that both investigations it constructed. tookplace at the sametime. The relationship was a two-wayroad as the thinking appliedto defining the Kao-fold inthe virtual space of the computer also helped to identifyways in whichto developthe various physical models. It wasdecided that the transformational qualities would be best modeled parametrically usingthe parametric modeling software, Catia. Two parametric models that would addresstwo different ways about thinking of the Kao-fold's transformation were devel- 32 oped.Interestingly, thelimitations ofthe software forced me to focus my thoughts such thatthey way in which the model was constructed would limit what transformations wouldbe possible inthe end. While a limit,one may argue that this, specifically, is thepower of parametricmodeling -to construct a digital 3d model in accordance with certainparameters orinherent characteristics ofthat form. 5. Herewe see all sketches in 3dspace. Bothmodels work with the Kao-fold in its half-arch configuration. The drawingofthe firstmodel was based on assigning anaxis to the dihedral angles and linking them to offsetplanes as well as to rotatingplanes. While the dimensions of the folded form wouldbe distorted, the intention ofthe model would be to playwith the angle of the dihedralangle axisand the distance between them to give morphings ofthe original geometry. Modelwas drawn referring to2-d sketches togive the 3-d coordinates ofthe corner points,which subsequently allows one to link them giving the seams of the folds. Thefollowing story board, illustrates the model's construction:

9. Allnodes or extremitiesof the two-dimen- sionalsketch are connected to give the trian- gulationof the pleated form. 4. Eachrotational A plane(which is linked h to theoffset planes) is associatedwith a two-dimensional sketch.Catia operatesbetween two workbenches- that of thesketch and the3d model. In the 'W' 641'.. +' . a 4 0 's . 4.. sketchmode, points thatwould later give projected... 3. ...whichwould function as a rotationalaxis 2.Extension lines were the3-d coordinates forthe rotational planes. forthe verticeswere drawn.

7. Everyrotational plane representsanaxis foreach dihedral angle.

T .LRZ

( I -7 j/T~ if i-i 1w / F'

4 !A ~-~+4 ~ '44. A -. I - I. C. Ok i I *. + 4 'k " ' A o " 4 4 4 1 ' _

12.The outlines are filled to givepanels of zerothickness. A

14.The first parameter, the distance between 15. theoffset planes,may beadjusted incremen- tallyto expandthe width of the half arch.

,4,0

19.... andthe camera angle may bechanged 21.Next, thesecond parameter, the position to givethe impression that the structure is of the rotationalplanes may be adjusted. lyingon its side......

51 -AIb@O~ftof&MAP

,~p ~p.4 ~ ------>

* i,~ ~17. 18.... orthe assignedoffset values may be randomized...

T Im. * IL

24.... andthen expanding 22.As with the offset planes, their assigned value may be 23. controlledand incremental, fistclosing in onone end...

%AUQ "Ak 4

25.... or randomand chaotic 26.If we play with both parameters, more exciting forms can be given. 27. Thesecond model was con- ceivedbased on the premise thatthe outer-most points of the formlay on the surface of a sphere.(see image below) Theparametric underpinning of themodel would be the function thatas the diameter ofthe sphereincreased, sodid the 1.I startedby using the dimensions from a 2.What was important about this dihedralangle of thefolded particularpattern was that its outer determined2-d crease pattern. "nodes"lying in the same plane also lie on a circleand that its inner form.And as the diameter nodeslie ona smaller,concentric circle in the same plane. approachedinfinity, the form wouldflatten out into a single planeor, if youcould bend yourimagination, a point on the sphere. ~iI 36 4;] I

6. ...allowingme to getthe intersections 7.Point definitions along thelength of the betweenthe sphere and the planes. archwere drawn taking the assigned extre- mumas the origin on each circle.

~~.4 j

brainstormingwith Alex on the chalkboard... 11.Along the x-axis or the width of the form, 12... whichtaper around an axis running each"collection" of pointslie on a circle... throughthe center of thesphere. 3. Beforeworking in Catia,a new2-d drawing to determine 5.These planes were subsequently trans- thedistance between the planes on which each and every ferredto Catia... dihedralangle lies had to bedeveloped.

WAp 1L4U4e

1 19

8. Eachpoint would reference the location of 9. Pointswere defined on either side of the 10.... giving us a "patch"of points. thevertices within the creasepattern. extremum...

#~A 4 . t

13.The distance between the points along the curveisidentical for 14.It was criticaltofix the distancebetween the points so 15.... regardlessof the diameterofthe sphere. eachpair ofpoints and taken from the determined2-dcrease pattern. thatthe 'patch" or areadefined by the pointson the sphere's surfacewould stay the same... Ai ~ ilti / U

16 Returningtothe original radius 17.... ineswere drawn by connecting the points... 18.... andfilled to createpanels,

22.We see the arch expand and unfold. 23.Finally, the radius is set to 1000ft,at which point the arch hasnearly folded flat. A

20.The final step wouldbeto assigna parametricfunction to 21.Subsequently, the assigned valueforthe theradius of the sphere that so as it increasedin size, the radiusis raisedto 100ft. "patch"or foldedform would flatten.The original radius is set at7'-10".

39 fabrication: beyonddiagrams & representations

Whileorigami may serve as an extremelyrich source of beautifulforms and whilemathematical reasoning, paper models, and digital simulations address interestingand useful methods for design, the bigquestion in mymind has alwaysbeen how to go beyonddiagrams, and abstract representations? The challengewould be to tacklethe complexity ofaddressing materials, components,and fabrication, as wellas otherreal-life issues, such as, howdoes this thingget transported and deployed?

40 Tobegin thinking of building in realityand building a detailed model, the first stepwould be to determinean application.Working within the anticipated time constraint,it wasdetermined that thestructure would be a small-scale,open- airshelter for a temporaryout-door event suchas an arts& crafts fair. The ensuingcriteria are that it beeasily transportable and easily deployed. More specificallyit:(a.) should be lightweight,(b.) may fold up andfit in the back ofa pick-uptruck, (c.) would be intuitiveand quick to assemble,that is, have a smallnumber of partsto assembleon site and assemble without the use ofa cranebut the labor of a fewpeople, and, (d.) should use non-precious materials,preferably reconstituted or recycled.As a deployablestructure, the formis notmanipulated or reconfiguredon the site but will be broughtthere as a prefabricatedassembly whereby the form and its corresponding folding patternwould be predeterminedafter a virtualexploration in the computer. Tothis end, fabrication would be baseddirectly on files generated from the computermodel, all components manufactured for a particularconfiguration of thefolded form. uggymuuuu: pos::uulli:::1g jjji::Im uii~i:;m gi~igi;;ipi::yyiiigii:iigg a~n::igli.siiii::::liiiiiia|M i-binii.-iiiiiii.. .iii.

41 Thecrease pattern and its form waschosen for its gentle cur- vaturewhich suggests a structure wherethewall and roof area continuoussurface provid- ingan adequate sense of enclo- sureand separating one booth orfunction from another, yet to notbe too enclosed, resulting in large areasofnon-habitable space.(Looking atthe previous collageofthe half-arch configu- rationon page 30, we can see howmuch space isgiven over to "storage".)I had decided against thearched form as from previ- 42 ouspaper models used to study habitationpossibilities and from thesimple materials experimen- tationmodels, I determined that thestructure would be most simplyerected on site if the seams werenotput under stress.One end of the shelter isconsiderably higher and out- stretched overtheother for sakeof orientation. It'sform is simpleallowing for easy repeti- tionin a larger,open area such asa parkor plaza. Usingthe same assembly asthe ply- woodmodel, the detailmodel recre- atesthe Kao fold using honeycomb boardwith canvas at the seams. Amockup construction using trash bagsdemonstrates thebasic assem- blyof two sheets of fabric pinned to andconnecting the panelsoneither side.(see image to right) Honeycomb boardwas selected for its recycled papercontent and high strength to weightratio. The honeycomb issuch anefficient construction that it is usedin trainbodies and airplane wings. (seeimage15,15a) 43 r Togive the structure lateral stability, aflexible yet relatively stiff mem- 2 branewould be required atthe (15a.) seams.As such, canvas was chosen assomething very affordable and (15.) available.Toconnect the canvas to thehoneycomb board, strips of 144# YUPO,a recycable, tear-resistant, syntheticpaper was used. Due to a shortsupply, 32nd" birch plywood wassubstituted onthe exterior. The canvaswould be riveted to the YUPOand birch strips, which, in turn,would be secured by a boltrun- ningthrough the panel.(see sketch) Modularbelts of differentthicknesses were foldedto testYUPO's flexibility. Usingthe dimensions ofthe honey-comb board soldas a limitation,the detail model was constructedat a scaleof 3/8thsof thefull scale.Furthermore, given material and time con- straints,the model focused on a portionof the entire structurethatcan be seen highlighted on thecrease pattern and marked in red on the chip-boardmodel. Withthis particular form, the structure essen- tiallybehaves like an accordion requiringa sec- ondaryarmature hold to the panels in place. Thisarmature would take the form of com- pressive strutsspanning vertically between one vertexand the next, theirlengths defining the respectivelocation of thevertices. Componentsthatwould receive the struts had 44 tobe designed. Given thetwist in thesurface ofthe overallstructure, the individual pairs of panelsand their respective struts do not align orthogonally.(see image-ortho) Thismeans the receivingends for the two struts meeting at a vertexwould haveslightly differing orientations andcan not function as a simplehinge (see imageprotoA).Toaccommodate forthis difference, one approachwouldbe to inserta gasket likematerial toallow for the strut to hingelater- allyas well as locally. (see image-protoB) Inthis model, Tyvek wasused at the seems con- nectingthe panels(see mockup assembly onpre- viouspage) and the hinge component isattached to anexcess portion of fabric at the vertex using grommets.(see image-protoC) image-ortho 45

image-protoA image-protoB image-protoC The problem this withmethod isthat a lotof stressis placed on the pin that con- nects thehinge to the strut such that it may easilyfatigue and fail under the structure's owndead weight. Adifferent approach was adoptedinwhich the hinge would connect directlyto the panelsand allow thearea of contactto pivot. (see imageto the right) Inorder to further rigidify the structure, the hingingaction would be limited to the struts whilethe component would hold the two panelsto which it wassecured at a fixed angle.This meant knowing each different anglefor everyvertex. Once the panelsof thechip-board model were crudely held in placewith brass struts and epoxy and its 46 final shaperesolved, the model was brought

tothe 3d-digitizer fora 3dscan givinga image-hingeA digitalmodel in Rhinoceros.Subsequently, the3d model wasimported into Catiaand bybisecting a plane through the vertex, their respectiveangles could be determined. Asa third layer of structural reinforcement, the panelsforming the dihedral angles wouldbe locked into place using a sec- ondarysystem of hinges.(seeimage-hingeA) These hingesarealso bolted directly to the panels.They are a simplehinge that may betightened once all hinges on the vertices, togetherwith the struts, are implemented together,and assembled......

1--E 19

Allcomponents were drawn in 13 14 AutoCADand designed tobe 17 ableto be laser-cut, glued 21 together,and assembled. 12

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* 'Na a Alsothe YUPO strips and the canvasjoints were drawn-up digitallyincluding holesforall boltsand rivets for precise assembly.(see previous page) Cuttingthe canvas was partic- ularlychallenging asit had tobe secured toa layerof chipboard in orderthat it not catchon fire. Furthermore, the dimensionsofthe laser bed led tosplicing separate pieces to createa continuousjoint. While speedandprecision were certainlyadvantages in using the lasercutter, it maybe 48 pointedout that the constraint imposedby the dimensionof thebed led to a largeamount of wastedmaterial.

1/8"thick, birch-faced plywood wascut in a singlepass. Thoughcut-lines were prepared forlaser cutting by nestling, a lot of materialwas wasted. Thefollowing thumbnail images form a storyboard whichhighlight the design process of going back andforth between the digital and the physical. WhatI learned was a designprocess. The model isa prototype,not an answer but the result of a chosendesign path with the toolsthat were givento me. While ease of deployment was a primeconcern with which I began the model, this objectivewas side-lined asthe structural response duringdeployment had not been given sufficient attention.As a consequenceofthe hinges break- ingunder too much bending stress and forcing themto delaminate, thecomponents could not be securedalone and an extra pair of hands was oftenneeded. Should the components have been machinedbya 3-axismilling machine out of steel, of course,this would not have happened. Yet, the 49 limitationsserved to guide the design and should theconnections beredesigned, they would notbe givena fixedangle but operate freely (as do the hingeson the interior) and the 3-dimensional loca- tionof eachvertex would only be dependent on the lengthofthe struts.This would take off a lotof pressure from the connections and enable oneto lockeach vertex in placestarting at the baseand working upwards. Overall, it was a very frustratingprocess to assemblethe structure but whatI learnedwas how often you lose a certain amountsofcontrol along the way due to various inaccuraciesandoverlooked difficulties and thus, theresult is an interpretation of the intentions with whichI began. ..

finiteelement analysis: an assessmentof structuralstability

Afterconstructing the detailed model, Iwanted to assess thedimensions of the components for structural stability as wellas testthe form for different material assemblies. Howwould the structurehold up under its own weight if constructedin full scale? Would it be ableto withstand lateralwind loads? Numerous simulationswererun in the finiteelement analysis program, SAP 2000. Different mate- rialswere tested under theirown dead weight and various windloads. In doing so, thethickness of theplates was Analysis_1:3/16" thick alupanel - 1/4"diaml.063"thick alupipe struts 52 variedand the diameter of the struts adjusted.

Unfortunately,thehoneycomb board could not be simu- latedand so thematerial that was closest in itsproperties, aluminum,was used. Moreover, given the time constraint, a Ti flexibleseam, as usedin theprototype, could not bemod- eledresulting in a rigidfolded-plate structure. Neverthe- less,it wasbelieved that testing the modelfor different loadconditions, albeit a differentconstruction, would give legitimacyto its form. Ibegan by running a simulationfor an aluminumstructure with3/16th inch thick plates, and a 1 "/.063"thick aluminum pipeto serveas thestruts. (see: Analysis_1) The displacement,measured at thehighest point was negligible.Next, a windload of 30psf was distributed over its entire 337 sq.ft. areagiving a total of10 kipsforce. (see: Analysis_2,2a) Analysis2:30psf wind load over 337 sq. ft. (10 kips) negativex -global Analysis_2a:30psf wind load over 337 sq. ft. (10kips) positive x -global Appliedboth to the concaveandconvex directionofthe surface,thedisplace- mentalong the axis of the wind load measure less than an inch.

Thetable lists displacement values disptaccnwnt. onlyunder its own dead load for 312 thick alupanel - 1dia/.63"hic aiuip stut d;spbccm# tiC anincreasing plate thickness and a dis plar ttiPJlt. 1"thick alupane! - 1"diam/ 063"thic aiupipe stut diminishingdiameter ofthe struts. displacetncnt. 2" thick aluparne - 1"dam/.063"tick aiupp stuts Asexpected, as the thickness ofthe plateswas increased, the greater o1o~disp/acement: 3/16" thick alupanel - 1"diam 0633"th/ickIauipe sturnts -.0d16~ displacemnent: 3/16" thick alu panel - 1/2"diamn .06 3"thick a'u-pie struts thedeflection, however, the change - 0431" displacemnent: 3/16" thick alupanel - 1/4"diam .063"Thic alPupip- strut wasnot linear given the greater - O52~J dispiacemient: 3/6" thick a/upane 1/8"daim &5"thick aupipe4struts rigidityobtained from the weld along theseams. Surprisingly, thestruts -17 dIs.placement: 3/ 16" thick a/u panei - no struts 1672 dispiacemnent: 3/16" thick stee/ - no struts couldbe reduced toa diameterof 1/8of an inch andthe displacement remainsunder half an inch. 0022" displacement: 172" concrete shiel 0022" displacement: 1 8" csneptesel Inrunning two separate sim- ulationsfor aluminum, while thescale factor exaggerates thedeflection, we see that the structurewould distort more uniformlywith struts on the inte- rioras well.

Imagininga more permanent application,stress diagrams wereobtained for a modelwith 3/16thinch thick steel plates showingthat it isgenerally stablebut would first show signsof fatigue at the folds in Analysis3:3/16th" steel plate - stressdiagram thebase where the deadweight 54 accumulates.

Finally, displacementreadings wereobtained for a full-arch configurationusing both steel andconcrete. They demon- stratethat, for either material, theKao-fold isa verystable structure.

Analysis_3a:3/16th" steel plate - stressdiagram Analysis_4:1/8" concrete shellw/-. 0022"displacement atsummit Analysis6: 1/2"concrete shell w/ -.0022"displacement at summit 55

Analysis_5:1/8" steel shell w/ -.0009"displacement atsummit conclusion:

WhatI learnedwas a designprocess; a process ofques- tioningthrough making. Though,atthe onset, certain goals werestated and kept in mind, each step would inform the nextallowing the designto evolve as if by a willof its own. Everyinvestigation, every toolhad offered certain insights, yetdue to various inaccuracies, the outcome alwayswas a transfigurationofthe original, or as mentioned, aninter- pretationofthe intentions with which I began. The results address thecreative potentialoforigami as a databaseof beautiful,sophisticated forms from which design ideas may beconceived. Should the exploration bepursued further, I wouldlike to test different pleated formsformaterial and assemblyideas with a revised approachtodetailing of 56 theconnections and of transportation, deployment, and dis- mantlingmechanisms. Forthe structure's overall optimiza- tion,the geometry and mechanism fordeployability should betailored more specifically to its application. Furtherfuture considerations include an evaluation ofthe formfor natural and artificial lightingconditions within and seenfrom the outside. From the inside, changing natural lightaddresses the notion of temporality byaltering one's perceptionofthe contained space. As such, the forms couldbe assessed for the nature of theirfacets and their abilityto create a rich andvaried distribution of light in responsetovarying sun angles throughout the day. Should thestructure be lit from within, at night,it's presence, much likea glowing lantern,cancelebrate a civic space forall toenjoy. '7

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.4 -'I comments&criticisms:

Whatfollows is a partialdocumentation of the comments and criticisms by my thesis committee and two invited guests, Patricia Patkau of Patkau andAssociates inVancouver and Hugh Stewart from Foster and Partners in London.

PatriciaPatkau: "Whatwas the biggest obstacle in making the jump over from paper to othermaterials?" K.G.: "Paperis extremely thin and therefore very malleable.. .itwill distort to permitcertain transformation aswell as to permitdistortion inthe final form...meaning certain folded forms would not be able to be folded from a singleplane starting point or at all if the unitswere made of stiff materialssuch as plywood orsheet metal. 58 PP: "Wereyou following one scale as you were working through the different models?" K.G.: "Yes,I determined this was a smallspace for a six-ft.person..30',15'."

HughStewart: "It'sa fantasticinvestigation and what it seemsyou've identified, for me, is that though origami is anattractive proposition atthe scale of paper models,it doesn't translate atthe scale of real-world building components... what you end up doing is to rigidify it byintroducing these struts... and arenot using the strength inthe folds, as is the caseinthe paper models. Ifthis is your area of interest, what I wouldsay you do is to identifywhat you are trying to achieveand then harness your analytical abilities to makethat happen... because you could go on forever... you need to identifya real task, a realproblem and then work backwards asto how youmeet the demands imposed by that problem..." K.G.: "Whenworking in a smallerscale, felt it wassomething I physically had a handleon.. .on a largerscale, I must admit that this was not an efficient deploymentofmaterial... I was constantly struggling with themodel..needed 3,4 people to helpme hold the nodes in placewhile I securedthe connections...hadto goback and forth between modifying what the computer told me was the solution to what realitypermitted.." JohnOchsendorf: "Thatis one of the key lessons learned -working between the two modes of digital and physical and what hit me was when you said you didn't wantto go back to the math.. .how you work between the two. I applaudmentioning the pile of waste material.

H.S.: "Doyou still have a fascinationfor origami... doyou still think there's an application? Asan inspiration for form - yes,but I'm less convinced about the construction aspects."

H.S.: "Strutsare hard to engineer.. .you could cover the surface with some splurge and rigidify it likespray-on concrete... the origami could serve asthe formwork."

P.P.: "Iappreciate the investigation and how clear you were about your intentions... might recommend you look more carefully atthe inhabitants... con- sider dailyliving... where do you put the furniture and so on?"

ShunKanda: 59 "Whatwill you doin the next three weeks?" K.G.: "Documentthe workand reflect on the processand its outcome... take a break!"

CarolBurns: "Ithink there is room for one more wringing out... and one thing I would recommend isthis evaluation oftools - oneincredible strength of project is thatyou have gone from your simple fascination ofthe sophistication of origami, you have bombarded your intuition by way of making with every kindof tool you could put in yourhands... and in thisschool there a lotof very different tools, right? So with your fabulous friends you made these fabulousforms... and your thesis could be about the limitsoftools available now or a methodof probingwith a varietyof tools...you talked about thedigital and the physical but thereissomething else that is very strong in your work which is the purely mathematical... which is neither digital norphysical... and I wouldsay that's an area on its own that's worth pursuing. I'mconvinced that you, Ken, could do 8 or 10more wringing exercises ...just to puta capstoneon this investigation..." J.O.: Justto wrap up you really covered a full range of things...you worked satisfactorily inthe digital world... tried to build them and I applaudyou for that...taught us a lotabout geometry.. .from this simple form came an astounding richness and complexity ofinvestigation. bibliography

BOOKS:

Marks,RobertW."Dymaxion World of BuckminsterFuller", Reinhold Publishing COrporation - NY, 1960 p.178,9 &p.206-8

Fuse,Tomoko"Unit Origami - Multidimensional Transformations", Japan Publications, Inc.- Tokyo2001

Escrig,F.& Brebbia, C.A. "Mobile and Rapidly Assembled Structures 2", Computational Mechanics Publications, Southampton 1996 p.63-71

Escrig,F.& Brebbia, C.A. "Mobile and Rapidly Assembled Structures 3", WIT Press, Southampton 2000 60 p.155-61

Gantes,C.J. "Deployable Structures: Analysis and Design", WIT Press, Southampton 2001 p.3-15,52-6,149

Pearce,Peter "Structure in Nature as a Strategyfor Design", MIT Press, Cambridge 1979 p.x-xv,24-8

JOURNALS:

Praxisvol. 1 "Deceit/fabrication officedA" p.10-13, "Interference Screen" p.24,5

A+UArchitecture &Urbanism July-Sept. 1984, p.67-72 CONFERENCEPROCEEDINGS:

Barreto,Paulo Taborda 1992 Lines Meeting on a SurfaceThe "Mars" Paperfolding. in Proc. 3rd Int. Meeting of OrigamiScience, Mathematics, andEducation p.342-59

DeFocatiis, D.S.A. &Guest, S.D. 2002 Deployable membranes designed from folding tree leaves Proc. R. Soc. Lond., BE/1-10 http://www- civ.eng.cam.ac.uk/dsl/leaves.pdf

Hall,Judy 1997 Teaching Origami to DevelopVisual/Spatial Perception. in Proc. 2nd Int. Conference onOrigami Science &Scientific Origami, p.279-91

Kobayashi,H.,Kresling, B. & Vincent, J.F.V. 1998 The geometry ofunfolding tree leaves. Proc. R. Soc. Lond., p.147-54 61 Kresling,Biruta 1996 Folded and unfolded nature. in Proc.2nd Int. Conference onOrigami Science &Scientific Origami, p.93-107

Ikegami,Ushio 2002 Iso-Area Polyhedra. in Proc. 3rd Int. Meeting of OrigamiScience, Mathematics, andEducation , p.20, 141

Sakoda,James Minoru 1992 Hikari-ori: Reflective Folding. in Proc.2nd Int. Conference onOrigami Science &Scientific Origami, p.333-41

THESES:

Liapi,Katherine A.2002 Transformable Architecture Inspired by the Origami Art: Computer Visualization asa Toolfor Form Exploration. University ofTexas WEBSITES:

http://www.uspto.gov/patftlindex.html

http://kdg.mit.edu/Matrix/matrix.html

http://theory.Ics.mit.edu/-edemaine/hypar/

http://www-civ.eng.cam.ac.uk/dsl/leaves.pdf

http://www.britishorigami.org.uk/theory/chen.html

62 http://www.greatbuildings.com/buildings/AirForceAcademyChapel.html

http://www.greatbuildings.com/architects/FelixCandela.html

http://www.bluffton.edu/-sullivanm/bigsail/bigsail.html

http://www.americansteelspan.com/storagesheds.html

http://www.yupo.com/

*Construction ToyStore "Construction Site" in Waltham- http://www.constructiontoys.com

http://images.google.com/images WNW

(1) http://ricegallery.org/view/exhibition/bambooroof.html

(2) http://www.Oll.com/lud/pages/architecture/archgallery/ito-serpentine/

(3),(4) Marks,RobertW."Dymaxion World of Buckminster Fuller'

(5) Escrig,F.& Brebbia, C.A. "Mobile and Rapidly Assembled Structures 2" imagecredits (6) A+UArchitecture &Urbanism July-Sept. 1984, p.67-72

* allimages are credited (7) Escrig,F.& Brebbia, C.A. "Mobile and Rapidly Assembled Structures 3" to theauthor unless listed inthis section (8,8a) Ikegami,Ushio 2002 Iso-Area Polyhedra

(9) DeFocatiis, D.S.A. &Guest, S.D. 2002 Deployable membranes designed from folding treeleaves p.9 63 (10) Fuse,Tomoko"Unit Origami - Multidimensional Transformations"

(11-11b) Gantes,C.J. "Deployable Structures: Analysis and Design" p.54,55

(12) Pearce,Peter "Structure in Nature as a Strategyfor Design" p.26,7

(13) http://www.bluffton.edu/-sulivanm/bigsail/bigsail.html

http://www.greatbuildings.com/architects/FelixCandela.html (14,14a)

(15,15a) http://images.google.com/images 64

(16.)