ebook img

Geometric Folding Algorithms: Linkages, Origami, Polyhedra PDF

487 Pages·2007·10.21 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Geometric Folding Algorithms: Linkages, Origami, Polyhedra

P1:FYX/FYX P2:FYX 0521857570pre CUNY758/Demaine 0521810957 February25,2007 7:5 GEOMETRIC FOLDING ALGORITHMS FoldingandunfoldingproblemshavebeenimplicitsinceAlbrechtDürer intheearly1500sbuthaveonlyrecentlybeenstudiedinthemathemat- icalliterature.Overthepastdecade,therehasbeenasurgeofinterest intheseproblems,withapplicationsrangingfromroboticstoprotein folding. With an emphasis on algorithmic or computational aspects, thiscomprehensivetreatmentofthegeometryoffoldingandunfolding presentshundredsofresultsandmorethan60unsolved“openprob- lems”tospurfurtherresearch. Theauthorscoverone-dimensional(1D)objects(linkages),2Dobjects (paper),and3Dobjects(polyhedra).AmongtheresultsinPartIisthat thereisaplanarlinkagethatcantraceoutanyalgebraiccurve,even“sign yourname.”PartIIfeaturesthe“fold-and-cut”algorithm,establishing thatanystraight-linedrawingonpapercanbefoldedsothatthecom- pletedrawingcanbecutoutwithonestraightscissorscut.InPartIII, readerswillseethatthe“Latincross”unfoldingofacubecanberefolded to23differentconvexpolyhedra. Aimedprimarilyatadvancedundergraduateandgraduatestudents inmathematicsorcomputerscience,thislavishlyillustratedbookwill fascinateabroadaudience,fromhighschoolstudentstoresearchers. ErikD.DemaineistheEstherandHaroldE.EdgertonProfessorofElec- tricalEngineeringandComputerScienceattheMassachusettsInstitute ofTechnology,wherehejoinedthefacultyin2001.Heistherecipientof severalawards,includingaMacArthurFellowship,aSloanFellowship, theHaroldE.EdgertonFacultyAchievementAward,theRuthandJoel SpiraAwardforDistinguishedTeaching,andtheNSERCDoctoralPrize. Hehaspublishedmorethan150paperswithmorethan150collabora- torsandcoeditedthebookTributetoaMathemagicianinhonorofthe influentialrecreationalmathematicianMartinGardner. Joseph O’Rourke is the Olin Professor of Computer Science at Smith CollegeandthefoundingChairoftheComputerScienceDepartment.He hasreceivedseveralgrantsandawards,includingaPresidentialYoung InvestigatorAward,aGuggenheimFellowship,andtheNSFDirector’s AwardforDistinguishedTeachingScholars.Hisresearchisinthefield ofcomputationalgeometry,wherehehaspublishedamonographand atextbook,andcoeditedtheHandbookofDiscreteandComputational Geometry. i P1:FYX/FYX P2:FYX 0521857570pre CUNY758/Demaine 0521810957 February25,2007 7:5 ii P1:FYX/FYX P2:FYX 0521857570pre CUNY758/Demaine 0521810957 February25,2007 7:5 Geometric Folding Algorithms Linkages, Origami, Polyhedra ERIK D. DEMAINE MassachusettsInstituteofTechnology JOSEPH O’ROURKE SmithCollege iii P1:FYX/FYX P2:FYX 0521857570pre CUNY758/Demaine 0521810957 February25,2007 7:5 cambridgeuniversitypress Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo CambridgeUniversityPress 32AvenueoftheAmericas,NewYork,NY10013-2473,USA www.cambridge.org Informationonthistitle:www.cambridge.org/9780521857574 (cid:2)C ErikD.Demaine,JosephO’Rourke2007 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. Firstpublished2007 PrintedintheUnitedStatesofAmerica AcatalogrecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloginginPublicationData Demaine,ErikD.,1981– Geometricfoldingalgorithms:linkages,origami,polyhedra/ErikD. Demaine,JosephO’Rourke. p. cm. Includesindex. ISBN-13:978-0-521-85757-4(hardback) ISBN-10:0-521-85757-0(hardback) 1.Polyhedra–Models. 2.Polyhedra–Dataprocessing. I.O’Rourke,Joseph. II.Title. QA491.D46 2007 516(cid:3).156–dc22 2006038156 ISBN 978-0-521-85757-4hardback CambridgeUniversityPresshasnoresponsibilityfor thepersistenceoraccuracyofURLsforexternalor third-partyInternetWebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuch Websitesis,orwillremain,accurateorappropriate. iv P1:FYX/FYX P2:FYX 0521857570pre CUNY758/Demaine 0521810957 February25,2007 7:5 Tomyfather,MartinDemaine Tomymother,EleanorO’Rourke –Erik –Joe v P1:FYX/FYX P2:FYX 0521857570pre CUNY758/Demaine 0521810957 February25,2007 7:5 vi P1:FYX/FYX P2:FYX 0521857570pre CUNY758/Demaine 0521810957 February25,2007 7:5 Contents Preface pagexi 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1 DesignProblems 1 0.2 FoldabilityQuestions 3 PartI. Linkages 1 ProblemClassificationandExamples. . . . . . . . . . . .9 1.1 Classification 10 1.2 Applications 11 2 UpperandLowerBounds . . . . . . . . . . . . . . . . . 17 2.1 GeneralAlgorithmsandUpperBounds 17 2.2 LowerBounds 22 3 PlanarLinkageMechanisms . . . . . . . . . . . . . . . . 29 3.1 Straight-lineLinkages 29 3.2 Kempe’sUniversalityTheorem 31 3.3 Hart’sInversor 40 4 RigidFrameworks . . . . . . . . . . . . . . . . . . . . . 43 4.1 BriefHistory 43 4.2 Rigidity 43 4.3 GenericRigidity 44 4.4 InfinitesimalRigidity 49 4.5 Tensegrities 53 4.6 PolyhedralLiftings 57 5 ReconfigurationofChains . . . . . . . . . . . . . . . . . 59 5.1 ReconfigurationPermittingIntersection 59 5.2 ReconfigurationinConfinedRegions 67 5.3 ReconfigurationwithoutSelf-Crossing 70 6 LockedChains . . . . . . . . . . . . . . . . . . . . . . . 86 6.1 Introduction 86 6.2 History 87 vii P1:FYX/FYX P2:FYX 0521857570pre CUNY758/Demaine 0521810957 February25,2007 7:5 viii Contents 6.3 LockedChainsin3D 88 6.4 NoLockedChainsin4D 92 6.5 LockedTreesin2D 94 6.6 NoLockedChainsin2D 96 6.7 AlgorithmsforUnlocking2DChains 105 6.8 InfinitesimallyLockedLinkagesin2D 113 6.9 3DPolygonswithaSimpleProjection 119 7 InterlockedChains . . . . . . . . . . . . . . . . . . . . 123 7.1 2-chains 125 7.2 3-chains 126 7.3 4-chains 127 8 Joint-ConstrainedMotion. . . . . . . . . . . . . . . . .131 8.1 Fixed-AngleLinkages 131 8.2 ConvexChains 143 9 ProteinFolding . . . . . . . . . . . . . . . . . . . . . . 148 9.1 ProduciblePolygonalProteinChains 148 9.2 ProbabilisticRoadmaps 154 9.3 HPModel 158 PartII. Paper 10 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 167 10.1 HistoryofOrigami 167 10.2 HistoryofOrigamiMathematics 168 10.3 Terminology 169 10.4 Overview 170 11 Foundations . . . . . . . . . . . . . . . . . . . . . . . . 172 11.1 Definitions:GettingStarted 172 11.2 Definitions:FoldedStatesof1DPaper 175 11.3 Definitions:FoldingMotionsof1DPaper 182 11.4 Definitions:FoldedStatesof2DPaper 183 11.5 Definitions:FoldingMotionsof2DPaper 187 11.6 FoldingMotionsExist 189 12 SimpleCreasePatterns . . . . . . . . . . . . . . . . . . 193 12.1 One-DimensionalFlatFoldings 193 12.2 Single-VertexCreasePatterns 198 12.3 ContinuousSingle-VertexFoldability 212 13 GeneralCreasePatterns. . . . . . . . . . . . . . . . . .214 13.1 LocalFlatFoldabilityisEasy 214 13.2 GlobalFlatFoldabilityisHard 217 14 MapFolding . . . . . . . . . . . . . . . . . . . . . . . . 224 14.1 SimpleFolds 225 14.2 RectangularMaps:Reductionto1D 227 14.3 HardnessofFoldingOrthogonalPolygons 228 14.4 OpenProblems 230 P1:FYX/FYX P2:FYX 0521857570pre CUNY758/Demaine 0521810957 February25,2007 7:5 Contents ix 15 SilhouettesandGiftWrapping . . . . . . . . . . . . . . 232 15.1 StripFolding 233 15.2 HamiltonianTriangulation 233 15.3 SeamPlacement 236 15.4 EfficientFoldings 237 16 TheTreeMethod. . . . . . . . . . . . . . . . . . . . . .240 16.1 OrigamiBases 240 16.2 UniaxialBases 242 16.3 EverythingisPossible 243 16.4 ActivePaths 244 16.5 ScaleOptimization 246 16.6 ConvexDecomposition 247 16.7 OverviewofFolding 249 16.8 UniversalMolecule 250 17 OneCompleteStraightCut . . . . . . . . . . . . . . . . 254 17.1 Straight-SkeletonMethod 256 17.2 Disk-PackingMethod 263 18 FlatteningPolyhedra . . . . . . . . . . . . . . . . . . . 279 18.1 ConnectiontoPartIII:ModelsofFolding 279 18.2 ConnectiontoFold-and-CutProblem 280 18.3 SolutionviaDiskPacking 281 18.4 PartialSolutionviaStraightSkeleton 281 19 GeometricConstructibility . . . . . . . . . . . . . . . . 285 19.1 Trisection 285 19.2 Huzita’sAxiomsandHatori’sAddition 285 19.3 ConstructibleNumbers 288 19.4 FoldingRegularPolygons 289 19.5 GeneralizingtheAxiomstoSolveAllPolynomials? 290 20 RigidOrigamiandCurvedCreases . . . . . . . . . . . . 292 20.1 FoldingPaperBags 292 20.2 CurvedSurfaceApproximation 293 20.3 DavidHuffman’sCurved-FoldsOrigami 296 PartIII. Polyhedra 21 IntroductionandOverview . . . . . . . . . . . . . . . . 299 21.1 Overview 299 21.2 Curvature 301 21.3 Gauss-BonnetTheorem 304 22 EdgeUnfoldingofPolyhedra . . . . . . . . . . . . . . . 306 22.1 Introduction 306 22.2 EvidenceforEdgeUnfoldings 312 22.3 EvidenceAgainstEdgeUnfoldings 313 22.4 UnfoldablePolyhedra 318 22.5 SpecialClassesofEdge-UnfoldablePolyhedra 321 22.6 Vertex-Unfoldings 333

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.