How to Fold It: The Mathematics of Linkages, Origami and Polyhedra

192 Pages·2011·6.589 MB·English

by  Joseph O'Rourke

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HOW TO FOLD IT
TheMathematicsofLinkages,Origami,andPolyhedra
Whatdoproteinsandpop-upcardshaveincommon?Howisopeninga
grocerybagdifferentfromopeningagiftbox?Howcanyoucutoutthe
lettersforawholewordallatoncewithonestraightscissorscut?How
manywaysaretheretoflattenacube?
Youcananswerthesequestionsandmorethroughthemathematics
offoldingandunfolding.Fromthisbook,youwilldiscovernewandold
mathematical theorems by folding paper and find out how to reason
towardproofs.
Withthehelpof200colorfigures,authorJosephO’Rourkeexplains
thesefascinatingfoldingproblemsstartingfromhighschoolalgebraand
geometryandintroducingmoreadvancedconceptsintangiblecontexts
as they arise. He shows how variations on these basic problems lead
directly to the frontiers of current mathematical research and offers
ten accessible unsolved problems for the enterprising reader. Before
tacklingthese,youcantestyourskillsonfiftyexerciseswithcomplete
solutions.
Thebook’swebsite,http://www.howtofoldit.org,hasdynamic
animations of many of the foldings and downloadable templates for
readerstofoldorcutout.
Joseph O’Rourke is Olin Professor and Chair of the Computer Sci-
ence Department, a Professor of Mathematics, and Director of Arts
and Technology at Smith College. His research is in computational
geometry,developingalgorithmsforgeometriccomputations.Hehas
wonseveralawards,includingaGuggenheimFellowshipin1987and
theNSFDirector’sAwardforDistinguishedTeachingScholarsin2001.
Hehaspublishedmorethan150papers,morethan30ofwhichwere
coauthoredwithundergraduates.Hehastaughtfoldingandunfolding
to students in grade school, middle school, high school, college, and
graduateschool,andtoteachers–ofgradeschool,middleschool,and
highschool–professors,andresearchers.Thisishissixthbook.

How to Fold It
The Mathematics of
Linkages, Origami, and Polyhedra
JOSEPH O’ROURKE
DepartmentofComputerScience,SmithCollege
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521145473
©JosephO’Rourke2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2011
PrintedinChinabyEverbest
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationData
ISBN978-0-521-76735-4Hardback
ISBN978-0-521-14547-3Paperback
Additionalresourcesforthispublicationathttp://www.howtofoldit.org
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsfor
externalorthird-partyInternetWebsitesreferredtointhispublicationanddoesnotguarantee
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Contents
Preface pageix
PARTI. Linkages 1
1 RobotArms 3
1.1 Annulus 5
1.2 ReachingAngles 15
1.3 Above&Beyond 20
2 Straight-LineLinkagesandthePantograph 24
2.1 Straight-LineLinkages 24
2.2 Pantograph 28
2.3 Above&Beyond 36
3 ProteinFoldingandPop-UpCards 39
3.1 Fixed-AngleChains 39
3.2 ProteinBackbones 40
3.3 MaximumSpan 42
3.4 Alignment 44
3.5 Piercing 46
3.6 Pop-UpSpinner 48
3.7 Above&Beyond 52
PARTII. Origami 55
4 FlatVertexFolds 57
4.1 MountainandValleyCreases 57
4.2 Single-VertexFlatFolds 58
4.3 TheMaekawa-JustinTheorem 61
4.4 TheLocalMinTheorem 64
4.5 TheKawasaki-JustinTheorem 66
4.6 Above&Beyond 68
v
vi Contents
5 FoldandOne-Cut 72
5.1 Examples 72
5.2 FoldandOne-CutTheorem 78
5.3 Above&Beyond 81
6 TheShoppingBagTheorem 84
6.1 TwoRigidOrigamiExamples 85
6.2 DihedralAngleConstraints 89
6.3 TheShoppingBagTheorem 93
6.4 Above&Beyond 96
PARTIII. Polyhedra 99
7 Dürer’sProblem:EdgeUnfolding 101
7.1 AlbrechtDürer’sNets 101
7.2 ConvexPolyhedra 103
7.3 TheOpenProblem 106
7.4 SpanningCutTree 109
7.5 SomePolyhedrawithNets 112
7.6 Above&Beyond 115
8 UnfoldingOrthogonalPolyhedra 119
8.1 OrthogonalPolyhedra 119
8.2 OrthogonalTerrains 120
8.3 GridUnfoldings 125
8.4 Above&Beyond 126
9 FoldingPolygonstoConvexPolyhedra 130
9.1 Questions 132
9.2 Alexandrov’sTheorem 133
9.3 FoldingConvexPolygons 135
9.4 TheFoldingsoftheLatinCross 138
9.5 Above&Beyond 140
10 FurtherReading 142
Glossary 147
AnswerstoExercises 151
Chapter1 151
Chapter2 155
Chapter3 156
Chapter4 158
Chapter5 161
Chapter6 162
Contents vii
Chapter7 165
Chapter8 168
Chapter9 170
Acknowledgments 173
Index 175