Table Of ContentOrigametry
Origami, the art of paper folding, has a rich mathematical theory. Early investiga-
tionsgobacktoatleastthe1930s,butthetwenty-firstcenturyhasseenaremarkable
blossoming of the mathematics of folding. Besides its use in describing origami
and designing new models, it is also finding real-world applications from building
nano-scalerobotstodeployinglargesolararraysinspace.
Written by a world expert on the subject, Origametry is the first complete reference
on the mathematics of origami. It brings together historical results, modern devel-
opments, and future directions into a cohesive whole. Over 180 figures illustrate
the material while numerous “diversions” provide jumping-off points for readers to
deepen their understanding. This book is an essential reference for researchers of
origamimathematicsanditsapplicationsinphysics,engineering,anddesign.Educa-
tors,students,andenthusiastswillalsofindmuchtoenjoyinthisfascinatingaccount
ofthemathematicsoffolding.
Origametry
Mathematical Methods in Paper Folding
THOMASC.HULL
WesternNewEnglandUniversity
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108478724
DOI:10.1017/9781108778633
(cid:2)c CambridgeUniversityPress2021
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2021
PrintedintheUnitedKingdombyTJInternationalLtd,PadstowCornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Hull,Thomas,1969–author.
Title:Origametry:mathematicalmethodsinpaperfolding/ThomasHull.
Description:NewYork:CambridgeUniversityPress,2020.|Includesindex.
Identifiers:LCCN2020022971(print)|LCCN2020022972(ebook)|
ISBN9781108478724(hardback)|ISBN9781108746113(ebook)
Subjects:LCSH:Origami–Design.|Origami–Mathematics.
Classification:LCCTT870.H852020(print)|LCCTT870(ebook)|DDC736/.982–dc23
LCrecordavailableathttps://lccn.loc.gov/2020022971
LCebookrecordavailableathttps://lccn.loc.gov/2020022972
ISBN978-1-108-47872-4Hardback
ISBN978-1-108-74611-3Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
This book is dedicated to Paul Chapman, who introduced me to origami
andgavememyfirstorigamibook.
Contents
Introduction page1
Part I Geometric Constructions 9
1 ExamplesandBasicFolds 11
1.1 ConstructinganEquilateralTriangle 11
1.2 DividingaSegmentinto1/nths 13
1.3 TrisectinganAngle 16
1.4 FoldingaRegularHeptagon 19
1.5 TheBasicOrigamiOperations 23
1.6 HistoricalRemarks 29
2 SolvingEquationsviaFolding 31
2.1 FoldingConics 31
2.2 SolvingSecondDegreeEquations 36
2.3 SolvingThirdDegreeEquations 38
2.4 CubicCurvesandBeloch’sFold 45
2.5 HistoricalRemarks 46
3 OrigamiAlgebra 48
3.1 DefinitionsandOrigamiArithmetic 48
3.2 TheFieldofOrigamiNumbers 50
3.3 OtherCharacterizationsofOrigamiNumbers 55
3.4 OtherWork:OrigamiRings 57
4 BeyondClassicOrigami 58
4.1 Multifolds 58
4.2 Lang’sAngleQuintisection 60
4.3 DescribingMultifoldOrigamiOperations 63
4.4 SolvingEquationswithMultifolds 66
4.5 ConstructionswithCurvedCreases 69
4.6 OpenProblems 71
viii Contents
Part II The Combinatorial Geometry of Flat Origami 73
5 FlatVertexFolds:LocalProperties 75
5.1 DefinitionsandFlat-Foldability 75
5.2 Mountain-ValleyParity 81
5.3 NecessaryandSufficientConditions 85
5.4 ConeFolds 88
5.5 CountingValidMountain-ValleyAssignments 90
5.6 TheConfigurationSpaceofFlatVertexFolds 96
5.7 MatrixModelforFlatVertexFolds 104
5.8 OpenProblems 105
5.9 HistoricalRemarks 105
6 Multiple-VertexFlatFolds:GlobalProperties 107
6.1 ImpossibleCreasePatterns 107
6.2 GeneralizedKawasaki:NecessaryConditionsandtheFoldingMap 110
6.3 GeneralizedMaekawa 112
6.4 Justin’sTheoremandPaperwithHoles 114
6.5 GlobalFlat-Foldability 119
6.6 Flat-FoldabilityIsNP-Hard 127
6.7 OpenProblems 135
7 CountingFlatFolds 137
7.1 Two-ColorableCreasePatterns 137
7.2 PhantomFoldsoftheMiura-ori 139
7.3 TheStamp-FoldingProblem 145
7.4 TetheredMembraneLatticeFolding 152
7.5 OpenProblems 157
8 OtherFlat-FoldingProblems 159
8.1 HowManyTimesCanWeFoldaSheetofPaper? 159
8.2 CanAnyShapeBeFoldedandUnfolded? 161
8.3 OrigamiDesign 165
8.4 TheRumpledRuble,orMargoulisNapkinProblem 172
8.5 TheFold-and-CutProblem 174
Part III Algebra, Topology, and Analysis in Origami 179
9 OrigamiHomomorphisms 181
9.1 SymmetryGroupsofFlatOrigami 181
9.2 ExamplesofOrigamiHomomorphisms 184
9.3 ApplicationstoOrigamiTessellations 186
9.4 OpenProblems 189
Contents ix
10 FoldingManifolds 191
10.1 IsometricFoldings 191
10.2 TheLocalStructureoftheSingularSet 193
10.3 Robertson’sTheorem 199
10.4 TheAngleSumandRecoveryTheorems 203
10.5 MaekawaandKawasakiforIsometricFoldings 205
10.6 OpenProblems 215
10.7 HistoricalRemarks 215
11 AnAnalyticApproachtoIsometricFoldings 217
11.1 LipschitzContinuousandRigidMaps 217
11.2 ApplicationstoDirichletProblems 223
Part IV Non-flat Folding 229
12 RigidOrigami 231
12.1 MatrixModelandNecessaryConditions 232
12.2 TheGaussMap 239
12.3 TheGaussMapandRigidOrigami 246
12.4 AnotherGeneralizationofMaekawa 251
12.5 OpenProblems 254
12.6 HistoricalRemarks 255
13 RigidFoldings 256
13.1 InfinitesimalRigidFoldability 256
13.2 AngleRelationshipsforSingle-VertexRigidFoldability 271
13.3 AnIntrinsicConditionforRigidVertices 280
13.4 OpenProblems 288
13.5 HistoricalRemarks 289
14 RigidOrigamiTheory 290
14.1 ComplexityofRigidFoldabilitywithOptionalCreases 290
14.2 ConfigurationSpacesofRigidFoldings 296
14.3 Self-Foldability 304
14.4 OpenProblems 317
References 319
Index 330
