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Folding Flat Crease Patterns with Thick Materials JasonS.Ku∗ FieldIntelligenceLaboratory 6 1 MassachusettsInstituteofTechnology 0 Cambridge,MA02139 2 Email: [email protected] n a J ErikD.Demaine 1 ComputerScienceandArtificialIntelligenceLaboratory 2 MassachusettsInstituteofTechnology Cambridge,MA02139 ] G Email: [email protected] C . s c [ ABSTRACT 1 Modelingfoldingsurfaceswithnonzerothicknessisofpracticalinterestformechanicalengineering.Thereare v manyexistingapproachesthataccountformaterialthicknessinfoldingapplications.Weproposeanewsystematic 7 andbroadlyapplicablealgorithmtotransformcertainflat-foldablecreasepatternsintonewcreasepatternswith 4 7 similar folded structure but with a facet-separated folded state. We provide conditions on input crease patterns 5 forthealgorithmtoproduceathickenedcreasepatternavoidinglocalselfintersection,andprovideboundsforthe 0 maximumthicknessthatthealgorithmcanproduceforagiveninput.Wedemonstratetheseresultsinparameterized . 1 numericalsimulationsandphysicalmodels. 0 6 1 Introduction : v Whilemuchoftheresearchincomputationalorigamiappliestofoldedsurfaceswithzerothickness(particularlystruc- i tures that fold flat), modeling folding surfaces with nonzero thickness is of practical interest for mechanical engineering. X Design approachesfor folding thick material have many varied applications from kinetic architecture [1] and solar panel r a deployment[2], torobotics[3]andnano-fabrication[4]. Theseapplicationshavemotivatedresearchintothemathematics andmechanicsofrigidlyfoldingthickmaterials[5,6,7]. Wediscusssomeoftheexistingtechniquesfortakingintoaccount materialthicknessinthefollowingsection. In this paper, we propose a new approachfor accommodatingthickness that modifies certain existing crease patterns intonewplanarfoldingpatterns,preservingsomestructureoftheoldcreasepatternwhilefoldingaformwhosefacetsare separated from one another in the final state. We describe a systematic and broadly applicable algorithm to transform an inputflat-foldablecreasepatternintoanewcreasepatternhavingafacet-separated,nearlyflatfoldedstate. Our approach for converting flat foldings into facet-separated foldings replaces each flat crease in the input crease patternbytwoparallelcreasessymmetricallyoffsetabouttheoriginalatadistanceproportionaltoanassignedcreasewidth satisfyingcertainpropertiesoftheoriginalcreasepattern. Insteadofonecreasefoldingflatwitha turnangleof180◦, the twonewcreaseshaveaturnangleof90◦. Thiscreasewideningcreatesdifficultiesatcrease-patternintersectionssincethe offsetcreasesnolongerconvergetoapoint. Materialinthevicinityaroundeachcrease-patternvertexisthusdiscardedto accommodatecreasewidening. Whilethismodificationcreatesholesinthematerial,itintroducesextradegreesoffreedom ∗Addressallcorrespondencetothisauthor. A B C D Fig.1. Someexistingthickfoldingtechniques:(A)HingeShift,(B)VolumeTrimming,(C)OffsetPanel,and(D)OffsetCrease. thatcanallowthewidenedcreasestofold. Additionallythealgorithmidentifiesandremovessomesurfacematerialonone sideofcreasestoavoidself-intersections. Weprovideconditionsoninputflatfoldedstatesforthealgorithmtoproduceathickenedcreasepatternavoidinglocal self intersection, namely that crease-pattern faces are convex and creases do not touch the insides of other creases in the input.Wealsoprovideboundsforthemaximumthicknessthatthealgorithmcanproduceforagiveninput.Wedemonstrate ourresultsinparameterizednumericalsimulationsandphysicalmodels. ExistingThickFoldingTechniques Therearemanyexistingapproachesthatseektoaccountformaterialthicknessinfoldingapplications,eachwiththeir ownstrengthsandweaknesses. Wediscussthetechniquesbelow,whicharealsoillustratedinorderinFigure1. A:HingeShift The hinge shift strategy shifts hinges out of plane to accommodate material thickness [8]. While readily useful in creatingone-dimensionalfoldingsofthickmaterial,thistechniqueishardertoapplyto2Dcrease-patternnetworks.Hinges startoutofplanesocannotbuildonexistingdesigntechniquesstartingfromacoplanarfoldingpattern.Inadditionfullrange offoldingmotionisrestricted. Arecentapproachextendstheideaofhingeshiftingtohigherdegreecreasepatternvertices, butthismethodisgeometricallyrestrictiveintheanglesandthicknessesallowed[9]. B:VolumeTrimming Thestrategypresentedin [1] trims the edgesof a thickenedsurfaceto overcomemanyof the difficultiesofthe hinge shifttechnique. However,thismethodalsosuffersfromdecreasedrangeofmotionandtheslantedsurfacescanbedifficult tofabricateinpractice. C:OffsetPanel Theoffsetpaneltechnique[10]isprobablythemostpromisinginapplicationbecauseitisveryflexiblewhileaccom- modating full range of motion. This method retains hinges at the folding plane but shifts the thick material away from thefoldingplane. While promising,fabricatingsuchstructurescan bedifficultrequiringrobuststandoffsto connectthick materialtohinges. X X L Fig.2. Fromlefttoright:(1)genericcreasepattern 0,(2)locallyflatfoldablecreasepattern withlayerorderinggraph ,(3)withreduced layerorderinggraphG ,and(4)flatfolding fX (X ). D:OffsetCrease In this paper we expand on the ideas presented in [11] which accommodatesmaterial thickness by widening creases withflexiblematerial,creatingahingefromatwo-dimensionalregionofmaterial. Weproposeamodificationoftheoffset crease technique that widens creases in a systematic way, replacing each crease with two ideal hinges withoutrelying on flexiblematerials.Whilethistechniquedoesnotpreserveexactstructureoftheinputcreasepattern,itcreatesastructurethat can be easier to fabricate than othertechniques. Additionally, the proposedtechniqueallows originalfacets to be parallel inbothflatandfoldedconfigurations,potentiallyallowingforalignmentofsurfacemountedcomponents. Wedescribethis techniqueindetailinthefollowingsections,concentratingfirstondefinitionsandthenthealgorithmitself. Related to the proposedmethod are a few other methodsfor accommodatingmaterialthickness. A patent by Hober- man[12]offsetscreasesinanon-parallelwaytoaccommodatethickness,butalsosuffersfromdecreaserangeoffoldangle anddoesnotnaturalhandlecreasepatternswithinternalvertices. Stillothermethodsinvolveaddingdegreesoffreedomby allowingfacestoslidelongitudinallyalongcreases,butcanbequitedifficulttofabricate[13]. Definitions Wewouldliketotakeasinputasurfacethathasbeenfoldedflatandoutputa“thickened”version. Inordertoperform thistask, wemustfirstspecifytheinputprecisely,namelytheflatfoldedstate. Wewilldescribeinputflatfoldedstatesby wayofcreasepatternsandnon-wrappinglayerorderinggraphs. LetacreasepatternX beafinitestraight-lineplanargraphembeddinginR2. Callcrease-patternedgesboundaryedges iftheyboundtheexteriorface,andcallthemcreasesotherwise. Similarly,callcrease-patternverticesexterioriftheybound theexteriorfacewithallotherverticesinterior. Whenwespeakofanglesaroundaninteriorvertexv,wearereferringtothe cyclicallyorderedsetofanglesbetweenadjacentedgesconnectedtov. Acreasepatternissaidtobelocallyflat-foldableif thealternatingsumofanglesaroundeveryinteriorvertexiszero.Asdiscussedlater,wewillalsorestrictlocallyflat-foldable creasepatternstohaveonlyconvexinteriorfaces. Certainlyif wearegivenasinputa flatfoldedsurface, thenetworkofcreasesontheunfoldedsurfacedefineacrease patternwhichwillbelocallyflatfoldable.Thenextthingtopindownistheorderingoflayersinthefoldedstate. Given a locally flat-foldable crease pattern X , a flat mapping function fX :X →R2 is a piecewise isometric mapping underwhicheachinteriorface ofX is congruent,interiorfacesthatsharean edgeinX sharethe sameedgein fX (X ), and exactlyoneofanytwoadjacentinteriorfacesin fX (X )isreflectedfromitsorientationinX (i.e.eachcreasehasbeenfolded). Thisfunctionuniquelyexistsforalocallyflat-foldablecreasepatternuptoisometry(seeFigure2). Hereweadapttheworkonlayerorderingpresentedin[14]. GivenanexistingflatfoldedsurfacewithcreasepatternX alayerorderinggraphL isadirectedgraphonthefacesofX withanedgebetweenfacesAandBifandonlyifthereexists somepointsa∈A and b∈B suchthat fX (a)= fX (b)(the facesoverlapin the folding). Thedirectionof the edgesin the directedgrapharegivenbyarbitrarilycallingthesurfacenormalofsomefaceintheflatfolding‘up’anddrawingedgesto pointtothefaceontopoftheother. Suchalayerorderingmaynotbewelldefinediffacesarenotconvex(partsofaface mayexistaboveandbelowanother);assuchwewillrestrictourselvestocreasepatternswithconvexfacesfortheremainder of the paper. Additionally, constructingthe desired face offset folded state will be impossible if the faces of the layering orderinggraphcontainsadirectedcyclebecausesomefacescouldnotbeordered. We willthusrestricttoonlyflatfolded surfaceswithacycliclayerorderinggraphswhosefacescanbepartiallyordered. Layerorderinggraphscanbeverycomplicated,typicallycontainingedgesontheorderofthesquarednumberofcrease- patternfaces. However,theyoftencontainsignificantredundancywithrespecttoprovidinglayerorderinginformation. For example,consideranedgeofalayerorderinggraph(A,B)fromcrease-patternfaceAtoB(BisontopofA),forwhichthere existssomeotherdirectedpathLfromAtoB. TransitivityensuresthatLenforcestheorderingconditionimposedby(A,B), so edge (A,B) is redundantand can be removed from the graph without losing any layer ordering information. We then implicitlyconstructthereducedlayerorderinggraphG fromthelayerorderinggraphL byidentifyinganysuchredundant edgeandremovingitfromthegraph.Thisprocessterminatesandresultsinauniqueoutputsinceitisatransitivereduction. Lastly,wedefineaflatfoldedstate(X ,G )asalocallyflat-foldablecreasepatterntogetherwithareducedlayerordering graph free from self-intersection. Specifically, for any crease x boundingfaces A and B and a third faceC which strictly intersectsx ,nodirectedpathexistsinthereducedlayerorderinggraphG betweenfacesAandBvisitsfaceC(faceCdoes notintersectcreasex ). Thisobjectwillserveastheinputtoourthickeningalgorithm. Notethataflat-foldedstateimpliesa creaseassignmenttoeachcrease(eithermountainorvalley)bycomparingtheorientationandorderoffacesaccordingtothe flatmappingfunction fX andG . Further,wecallthereflexsideofacreasedsurfacetheoutsideofthecrease,andsimilarly wecalltheconvexsideofacreasedsurfacetheinsideofthecrease. A restriction on our approach is if two creases in a crease pattern wrap around each other in the flat folded state, specificallyifonecreasetouchestheinsideofanothercrease,selfintersectioncanbecomeaproblem.Wewillgointomore detailaswedescribethealgorithm,butfornowwewillcallaninputflatfoldedstatenon-wrappingifnocreaseorboundary edgepointoftheinputtouchestheinsideofanothercrease. Algorithm Thegoalofthispaperistoconstructathickenedversionofagivenanon-wrappingflatfoldedstate(X ,G ). Thestrategy is to offset crease-pattern faces from their flat folded state consistent with their layer ordering and create new creases to accommodatetheoffset.First,wemustdefineanoffsetdistancebetweeneverypairoffaceswhichimpliesawidthforeach crease. Second,weconstructscalablepolygonsateachinteriorcrease-patternvertexfromwhichmaterialwillberemoved toaccommodatewidenedcreases. Third,werefinethepolygonstoensurethateacheffectivevertexdoesnotexhibitlocal selfintersection. Fourth,wecalculatearangeforallowablescalefactorssuchthatvertexpolygonsdonotintersect. Fifth, welayoutthenewcreasepatternwithholeshavinganon-flatfoldedstateaccordingtoachosenscaleintheallowablerange. Last, we addressconstructingthe thicknessof eachface basedonavoidinglocalself intersection. Additionaladjustments maythenbemadetoaccountforglobalselfintersections. Step1: CreaseWidth The first goal of the algorithm is to specify a width for each crease in a flat folded state (X ,G ), with all mutually consistentwiththelayeringorderofoffsetfaces. Intuitively,wewanttoseparatethelayersoftheinputbynonzeroamounts andassignacreasewidthbasedonthedistancebetweenadjacentfaces. Ifcreasewidthsarechosensmall,wecanthinkof thedesiredoutputasan“almostflat”versionoftheoriginalthatallowsfornonzerospacebetweenlayers. Theconceptof creasewidthisrelatedtothesametermappliedtotheone-dimensionalstampfoldingproblem[15],butweapplyitto2D flat-foldablecreasepatternswithsortablelayerorderings.Forourpurposes,givenreducedlayerorderinggraphG itsuffices tochooseapositiveweightforeachdirectededgesuchthatgivenanytwointeriorcrease-patternfacesAandB,everypath fromAtoBinG hasthesameweightsum. Wewillcallsuchaweightassignmentw :x ∈X →R+. Sucha weightassignmentalways exists; particularlyonecan be constructedbychoosingan arbitrarylinearizationof thepartialorderprescribedbyG tocreateatotalorder,anddefiningtheweightalongacreasetobetheabsolutedifference betweenthelayerorderingnumbersofthecrease’sincidentfaces. BygivingaweighttoeachcreaseofG ,wecancalculate acreasewidthforeverycreaseofX bysummingthetotalweightalonganypathfromonefaceincidenttothecrease,tothe other. Thechoiceofw canbeviewedasadesignchoiceforthealgorithmimplementer. Onemightstrivetochooseanw that optimizessomenaturalmetricsuchasminimizingthemaximumthicknessofanycrease,buttheworkin[15]and[16]seem to suggest such questionsmay be NP-hard even for one-dimensionalgraphs. As such, we do not attempt to optimize the choiceofw here,andleavetheexplorationinthisareaasanopenproblem. Oncewehaveassignedacreasewidthtoeachcrease,theconstructioninvolvesreplacingeachcreaseintheinputcrease patternwithtwoparallelcreasessymmetricallyoffsetabouttheoriginal,separatedatadistanceproportionaltotheassigned A u w p v Fig.3. Polygon construction. A genericinternal crease pattern Fig.4. Anon-simple vertex polygon andrefinement byclipping vertexshowingrelationshipbetweenoffsetsandangles. crossings. creasewidth. Thisreplacementcreatesdifficultiesatcreaseintersectionssincetheoffsetcreaseswillnolongerconvergeto a point. Materialin the vicinityaroundeachcrease-patternvertexwillneed to bediscardedto accommodatethe widened creases. Next,wewilldiscusstheconstructionoftheregiontobediscarded. Step2: PolygonConstruction Nowthatcreasewidthshavebeendefined,wemustinterfacewidenedcreaseswitheachotherinthevicinityofcrease- pattern vertices. For each vertex, we constructa polygonthat will interface with widened crease lines aroundthe vertex. Thesepolygonswillbescalablebasedonhowthickwewouldliketomakethematerialwithrespecttothecreasepattern,up toapoint. Wewilldealwiththeallowablerangeofscalingfactorlater. First,wemustdefinethegeometryofthesevertex polygonssotheywillalignwithallthecreasewidthsaroundthevertex. We wanta vertexpolygonto containonevertexper faceadjacentto the crease-patternvertexat a distancefromeach adjacentcrease proportionalto the crease width ofthe crease. Consider crease-patternvertexv with face A adjacentto it, boundedbyadjacentcreases{u,v}and{v,w}withcreasewidths2aand2brespectively.Lettheanglebetweenthesecreases be q . Then the location of the polygonvertex p in this face must be a distance a from crease {u,v} and distance b from crease{v,w}. Thispointisuniquelydefinedandcanbeparameterizedbythelengthh ofsegment{v,p}andtheanglesa andb betweenthissegmentandcreases{u,v}and{v,w}respectively(seeFigure3). Sometrigonometryrevealsthatthese anglesaregivenby sinq sinq tana = , tanb = (1) b/a+cosq a/b+cosq withdomainsa ,b ∈[0,p ],andh=a/sina =b/sinb . Repeatingthisprocedureforeachfaceadjacenttoaninteriorcrease- patternvertexconstructspointsthatwhenconnectedbasedonfacetadjacencyforma polygon. Forexteriorcrease-pattern vertices,thesameconstructionappliesexceptweincludetheoriginalvertexandintersectionsbetweencreasewidthlinesand boundaryedgesinourpolygons.Wecalltheregionsineachfaceboundedbyoffsetcreasesreducedfaces(showninbluein thefigures).Unfortunately,edgesofaconstructedvertexpolygonmayproperlycrossasinFigure4. However,wecaneasily modifythe vertexpolygonto be weakly simple, or evenconvex,byclippingany facetsector crossingthe polygon. More specificallytakingtheconvexhullofthevertexpolygon,markeachvertexwhoseadjacentreducedfacedoesnotproperly intersecttheconvexhull. Trimmingtheintersectingreducedfacesagainsttheconvexhullofthemarkedverticesresultsin anappropriateconvexvertexpolygon,thoughinsomecasesitmaysufficetoremovelessmaterial(seethemiddlediagram inFigure4). Notethatcrossingscanonlyariseiftwoadjacentvertexanglessumtomorethan180◦. Thereducedfacesof thesetwoanglescannotproperlyintersecttheconvexhullofthevertexpolygon,soatleasttwomarkedverticesexist. Locally, this polygon divides the area around the vertex into three region types: the polygon, widened creases, and reducedfaces(thecardinalityofthelattertwoequalingthenumberofcreasesadjacenttothecrease-patternvertex).Wewill usethisterminologytotalkabouttheseregionsinthefollowingsections. Step3: Refinement The newly constructed creases and polygonsin the previoussections serve to locally satisfy isometry between offset faces by removing material at a vertex and adding new creases to accommodatethe offset. However, creases with larger Fig.5. Trimmingintersectingregion. Fig.6. Unboundedintersectionforinsidetouchingcreasesinin- putflatfoldedstate. Fig.7. Scalefactorcalculationshowingrelevantquantities. creasewidthrequiremorepapertobeabsorbedintowidenedcreaseregions,reducingthesizeofsurroundingreducedfacets. Theinteractionofthistradeoffbetweendifferentregionscreatesthepotentialforintersectionbetweenwidenedcreasesand reducedfacets. Wefixthistypeofselfintersectionbycheckingeachwidenedcrease/reducedfacetpairforintersection. If theyintersect,trimthereducedfacetalongthewidenedcreaseboundaryandrefinethevertexpolygontoreflectthischange (seeFigure5). Thereisaworrythatthisprocedurecouldremovematerialthatisnotaboundeddistancefromthevertex.Forexample, the crease pattern shownin Figure 6 containstwo creasesthatwhen widenedhave an intersectionthat extendsto infinity. Fortunately,this type of situation onlyoccurslocally when some crease of the inputtouchesthe inside of anothercrease, whichwehaveforbiddenbyrequiringanon-wrappinginput. Reducedfacetscanonlybetrimmedafinitenumberoftimes becausetrimmingcannotincreasethenumberofintersections,thustherefinementterminates. Step4: ScaleFactor Aftercreatingvertexpolygonsandlocalwidenedcrease/reducedfacetregionsthatlocallydonotselfintersect,wecan determinehowlargethesepolygonscanbebeforeintersectingeachother. Eachwidenedcreaseedgeisboundedoneither sidebyavertexpolygon.Considercreasex withlengthisd.Theneachwidenedcreaseedgeofx isshorterthandaccording to the size of each incident vertex polygon. Let (h ,a ) and (h ,b ) define the locations of the vertex polygonvertices on a b eithersideofx containedinthesamefaceF. Ifweletthesizeofallvertexpolygonsscalebyafactors,thenthelengthℓx ofthewidenedcreasesegmentinF isgivenbythefollowingfunctionofs(seeFigure7): ℓx (s)=d−s(hacosa +hbcosb ). (2) For (hacosa +hbcosb ) negative, ℓx (s)>0 for all s>0 so this crease x does not restrict scale. For (hacosa +hbcosb ) positive,thereexistssomesx strictlypositiveforwhichℓx (sx )=0. Thiseventcorrespondstoneighboringvertexpolygons intersectingwhich we wouldlike to forbid. Taking the minimumsx overall creases x ∈X yieldsa strictly positive upper bounds∗ onscale factorsbywhichvertexpolygonscan bescaled withoutoverlap. Notethatfors=0, thecrease pattern isnotoffsetatallandfacetsremaincoplanar,andthefoldedformcannotbeproducedwithmaterialofanyfinitethickness. Strictly positive s, such as s∗ calculated above, allow the modified pattern to accommodate some finite thickness, with a largersaccommodatingalargerthicknessrelativetothegeometryoftheinputcreasepattern. Ofcoursethisscales∗ only insuresthatvertexpolygonsdonotinteract.Itispossiblethatwiththiscalculatedscale,globalintersectionbetweenfacesof thefoldingcanstillarise. Nonetheless,weshowthefollowing: Theorem1. Given a non-wrapping flat folded state (X ,G ) and weight assignment w :x ∈X →R+, there exists some positive non-zero scale s for which the above construction globally contains no strict intersection between faces in the three-dimensionalfoldedstate. Proof. Supposeforcontradictionthattheconstructionproducesintersectingfacesforeverypositivenon-zeroscales. Inter- section cannotoccurbetween reducedpolygonfaces becausethey are offsetfromeach otherin a way consistentwith the inputnon-wrappingreducedlayerorderinggraphcontainingnoself-intersection.Thus,anyface-faceintersectionmustexist betweenawidenedcreasefaceandsomeotherface. Letx betheoriginalcreasecorrespondingtosomewidenedcreaseface strictly interestinganotherface F. Because the inputis non-wrappingcontainingnoself-intersection, x doesnotintersect F inthe inputnon-wrappingflatfoldedstate. Increasingthe scale s from0 andperformingtheaboveconstructionresults in a continuousparameterizedfamily of three-dimensionalfoldings. More importantly,let d(s) be the minimum distance betweenthe widenedcrease associated with crease x in a constructionwith scale s and the reducedpolygonformedfrom faceF. Thend(s) is positivefor s=0 andvariescontinuouslyandweaklymonotonicallywith s. Thusthereexistssome positivenon-zeroscales′∈(0,s∗)forwhichd(s′)isalsopositive. Butthisistrueforeveryintersectioninvolvingacrease, sotheremustexistsomescalewherenointersectionoccur,acontradiction. (cid:3) Step5: FinalConstruction Now given a flat folded state (X ,G ) and width assignment w , we can calculate the upper bounds∗ on scale to forbid vertexpolygonintersectionandchooseascale s′ intherange(0,s∗)toconstructamodifiedcreasepatternthatavoidsself intersection. Quitesimplytheconstructionisplacingvertexpolygonsscaledbys′ andaddingwidenedcreaselinesparallel totheoriginalcreasesbetweenvertexpolygons. TheentireprocessisshowninFigure8: firsttheinputnon-wrappingflat foldedstate,offsetfacets,andfinallytheoffsetpolygons,togetherwiththeircounterpartsinthefoldingdomain. Theorem2. Givenanon-wrappingflatfoldedstate(X ,G )andweightassignmentw :x ∈X →R+,theconstructionabove terminatesinpolynomialtime. Proof. Giventheweightassignment,thevertexpolygonsareeachbounded,constructedasdescribedinStep2byoffsetting theoriginalgeometrybyfiniteamountsandconnectingvertices,whichcanbeconstructeddirectlyinlineartime. Clipping ensuresthevertexpolygonsareweaklysimpleandcanbeperformednaivelybycomparingeachvertex-edgepairinquadratic time. TrimminginStep3canalsobeimplementedinquadratictimebycheckingeachpairoffaceslocallyaroundavertex. Local intersections of the faces around a vertex are thus removed in the trimming step by construction. Calculating the scaleupperbounds∗ guaranteeingthatvertexpolygonsdonotintersectrequiresaconstant-sizedevaluationperedge,while calculating an appropriates′ can be computedby evaluating the appropriatescale for each possible intersection pair in at mostquadratictime, and choosingthe minimumscale. Thusthe procedurecan be implementedto terminate in quadratic timewhichispolynomial. (cid:3) AddingThickness The above construction creates a modified thin crease pattern that separates faces in the folded form to make room forthickpanels. Addingmaterialtothe constructedthinsurfaceis relativelyeasy. Ingeneral,if creasewidthsarechosen arbitrarily, facets can be assigned a range of thicknesses to either side that can be accommodated by the crease widths. However,asimplerandmorepracticalassignmentmightbetoassignthesamemaxthicknesstotheentirecreasepatternas manymanufacturingprocessescouldbenefitfromthiskindofuniformity(nano-fabrication,sheetmetalconstruction,etc.). Wecansimplydefinethemaxpanelthicknesst asthesmallestcreasewidthassignedtotheflatfoldedstate. max However,thispanelthicknesscannotbeaddedeverywhereormaterialwouldself-intersect. Forexample,iffinitepanel thicknessexistseverywhereonadjacentfacesontheinsidevalleysideofeachcrease,thecreasewouldnotbeabletofoldat arightanglewithouttheaddedmaterialintersectingwhenfolded. Therearemanywaystosolvethisproblembyremoving Fig.8. Constructionprocess. Fig.9. Foamcoremodelofathickenedtraditionalbirdbase. material.Wesuggestkeepingfullpanelthicknessonwidenedcreaseregionstostrengthenthesetraditionallyweakinterfaces, andremovingmaterialfromthe adjacentface incidentto the crease. To accommodatewidenedcrease panelthicknesson bothsides,wemustremoveastripofmaterialofwidtht /2oneithersideofthewidenedcreasefromthereducedfacets max adjacent to the crease, only on the crease’s inside surface. This modification will ensure that material in the vicinity of creasesdonotlocallyself-intersect. Theproblemofglobalmaterialselfintersectionduringafoldingmotionisamoredifficultcomputationaltask,though thereare existing computationalmethodsfor addressingthisissue. Theoffsetpaneltechniquesof [10] also pointoutthis problem.Wearelookingintomoreefficienttechniquestoperformglobalfoldingmotioncollisiondetectiontoaidreal-world designapplications. Models We developednumerical and physical models to demonstrate the algorithm presented above. We used the algorithm describedto modifytwo existingrigid-foldableflat-foldablecrease patterns, the traditionalbird base and a modifiedrigid foldableflappingbirddesignedbyRobertLangasshowninFigure10(birdbaseontop,andflappingbirdonbottom). On the left the originaland modified crease patternsare shown, followed by snapshotsof each crease pattern folding. These modifiedcreasepatternswereinputintoanumericalorigamisimulatorcalledFreeformOrigami[17]. Thissimulatorisable to fold a crease pattern incrementally through rigid folding configuration space while seeking to maintain developability and planarity constraints convergingiteratively to within double precision. Folding these crease patterns in the simulator demonstratedmultiplerigidfoldedstatesthroughoutthefoldingprocesstoveryhighaccuracy. Thesesimulationsprovide evidence that a path through the configuration space exist for complex crease patterns between the unfolded and folded statesproducedbythisalgorithm.Suchamovementseemspossibleforsingle-vertexcreasepatternsbecausethenumberof a12 0.785398 a23 1.5708 rho 2.09858 l1 0.5 l2 0.692 l3 0.724 l4 0.012 l1 0 0.5 1 l2 0 0.5 1 l3 0 0.5 1 l4 0 0.5 1 Fig.10. Numericalfolding simulationoftwo thickenedcrease patterns Fig.11. Parameterizedthicksinglevertexconstructionin usingFreeformOrigami. Mathematica. degreesoffreedomofthemodifiedstructureshouldingeneralincrease. We also used a Mathematica model shown in Figure 11 to apply the algorithm to single-vertexcrease patterns to try andfindapathinthefoldingconfigurationspacebetweentheunfoldedandfoldedstatesproducedbythisalgorithm. The model allows the user to change the parameters of the system, namely fold angles between creases and splitting ratios betweenoffsetcreasepairs,inordertosatisfyclosure. Whilewehavenotfoundananalyticclosureconstraintrelatingfold anglesandsplittingratios,wehavebeenabletoachieveclosurenumericallytodoubleprecisionforarangeofinputs. Our resultsinthisareaarepreliminary,butwehaveexperimentalevidencetosupportthatsingle-vertexcreasepatternsthickened with this algorithm have a rigid foldable path between unfoldedand folded states. We conjecture that the state space for thickened single-vertex crease patterns is a sphere embeddedin the multidimensionalparameterized space and will leave furtherdiscussioninthisareatofuturework. Lastly,aphysicalmodelofathickenedversionofthetraditionalbirdbasewasfabricatedusing3/8”foamcorepasted oneithersideofthinpaper. SomeviewsofthephysicalmodelcanbeseeninFigure9. Thefoldingactionobservedwith this modelagreeswell with the foldedstates of numericalsimulation, and the motionfeels tightlyconstrainedin contrast to the folding mechanisms described in [11]. Empirically fixing the dihedral angle between sector faces while adjusting theangleratioatonecrease,acontinuousadjustmentoftheothercreaseratioswasobserved,alsosupportingthespherical configurationspaceconjecture. Conclusion Inthispaperwehavepresentedanewmethodforcreatingthickfoldedstructuresfromflatfoldedstates. Thealgorithm proposedhasmanybenefitsoverexistingthickfoldingtechniques. Facetsurfacesintheproducedstructure’sunfoldedstate are coplanar allowing for ease of fabricationin layer-by-layermanufacturingprocesses. These same surfaces are parallel in the produced structure’s folded state allowing any surface mounted components to mate naturally. Panel thicknesses can be adjusted according to material and scale within bounds provided by the algorithm. Further, every finite area of the algorithm’s produced surface may be assigned non-zero thickness, allowing for the production of strong and tightly constrainedmechanisms. Theoffsetcreasemethodprovidesathickenedfoldedstatesuggestingafullrangeoffoldingmotionaswellasparallel facets when fully folded. Assigning crease widths to comply with the acyclic layer orderingof the input flat foldedstate providesaflexibledesignspaceforvariedapplications,whilestillconstructingonenon-trivialfoldedstatewithplanarfacets. Whileitisstillopenwhetherapathofrigidfoldedstatesexiststhroughtheconfigurationspaceingeneral,thereisevidence thatoneexistsforsinglevertexcreasepatternsgivenournumericalmodels. Whilecompensatingformaterialthicknessis notasdifficultfornon-flatfoldings,becausefacesdonotmeeteachotherwhenfolded,weareexploringwaysofextending thismethodfornon-flatfoldings,particularlythosecontainingface-to-facecontactintheirnon-flatfoldedform. Acknowledgements SupportedinpartbyNSFODISSEIgrantEFRI-1240383andNSFExpeditiongrantCCF-1138967. Wealsothankour anonymousreviewersforhelpfulcommentsandadditionalreferences. References [1] Tachi,T.,2011. “Rigid-foldablethickorigami”. Origami5,pp.253–264. [2] Schenk, M., Kerr, S., Smyth, A., and Guest, S., 2013. “Inflatable cylinders for deployable space structures”. In Proceedingsofthe1stInternationalConferenceTransformables2013. [3] Balkcom,D.,2002. “RoboticOrigamiFolding”. PhDThesis,CarnegieMellonUniversity,Pittsburgh,PA,August. [4] Arora, W. J., In, H. J., Buchner, T., Yang, S., Smith, H. I., and Barbastathis, G., 2006. “Nanostructuredorigami3d fabrication and self assembly process for soldier combat systems”. 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