Main
Twists, tilings, and tessellations : mathematical methods for geometric origami
Twists, tilings, and tessellations : mathematical methods for geometric origami
Lang, Robert J. (Robert James) 1961 author
0 /
0
How much do you like this book?
What’s the quality of the file?
Download the book for quality assessment
What’s the quality of the downloaded files?
Year:
2018
Publisher:
CRC Press
Language:
english
Pages:
757
ISBN 10:
148226241X
ISBN 13:
9781568812328
File:
PDF, 406.19 MB
Download (pdf, 406.19 MB)
 Open in Browser
 Checking other formats...
 Please login to your account first

Need help? Please read our short guide how to send a book to Kindle
The file will be sent to your email address. It may take up to 15 minutes before you receive it.
The file will be sent to your Kindle account. It may takes up to 15 minutes before you received it.
Please note: you need to verify every book you want to send to your Kindle. Check your mailbox for the verification email from Amazon Kindle.
Please note: you need to verify every book you want to send to your Kindle. Check your mailbox for the verification email from Amazon Kindle.
You may be interested in Powered by Rec2Me
Most frequently terms
crease^{1587}
vertex^{1287}
fold^{1080}
angles^{1029}
crease pattern^{872}
vertices^{794}
tiles^{555}
polygon^{554}
folds^{514}
sector^{493}
tiling^{475}
creases^{473}
origami^{469}
tile^{414}
graph^{405}
foldable^{397}
twists^{382}
tessellation^{348}
tessellations^{302}
cos^{294}
assignment^{273}
dual^{267}
fold angle^{265}
polygons^{256}
tilings^{245}
fold angles^{241}
sector angles^{231}
rotation^{228}
triangle^{224}
offset^{221}
pleat^{220}
sphere^{218}
pleats^{217}
valley^{213}
miura^{212}
twist tiles^{204}
equation^{202}
cyclic^{198}
tan^{193}
primal^{192}
folding^{191}
periodicity^{191}
crease assignment^{182}
vector^{175}
foldability^{169}
spherical^{167}
theorem^{165}
dual graph^{165}
assignments^{164}
facets^{161}
periodic^{161}
central polygon^{158}
ori^{158}
construct^{156}
vectors^{152}
rotational^{140}
crease patterns^{138}
cot^{135}
offset twist^{135}
symmetry^{134}
0 comments
You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.
1

2

Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 334872742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acidfree paper International Standard Book Number13: 9781138563063 (Hardback) International Standard Book Number13: 9781568812328 (Paperback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 9787508400. CCC is a notforprofit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may ; be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress CataloginginPublication Data Names: Lang, Robert J. (Robert James), 1961 author. Title: Twists, tilings, and tessellations / Robert J. Lang. Description: Boca Raton : CRC Press, 2018. Identifiers: LCCN 2017030497  ISBN 9781568812328 (pbk.) Subjects: LCSH: Combinatorial designs and configurations.  Twist mappings (Mathematics)  Tiling (Mathematics)  Tessellations (Mathematics)  OrigamiMathematics. Classification: LCC QA166.8 .L36 2018  DDC 516/.132dc23 LC record available at https://lccn.loc.gov/2017030497 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To Klaus Peters Contents Introduction 1 2 Genesis ? . . . . . . . . . . . . . . . . . . . . What to Expect and What You Need ? . . . . . 1 Vertices 1.1 1.2 1.3 1.4 1.5 Modeling Origami ? . . . . 1.1.1 Crease Patterns ? . . 1.1.2 Creases and Folds ? . Vertices ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 KawasakiJustin Theorem ? . . . . 1.2.2 Justin Ordering Conditions ? . . . . 1.2.3 Three Facet Theorem ? . . . . . . . 1.2.4 BigLittleBig Angle Theorem ? . . 1.2.5 MaekawaJustin Theorem ? . . . . 1.2.6 Vertex Type ? . . . . . . . . . . . . 1.2.7 Vertex Validity ? . . . . . . . . . . Degree2 Vertices ? . . . . . . . . . . . . . Degree4 Vertices ? . . . . . . . . . . . . . 1.4.1 Unique Smallest Sector ? . . . . . . 1.4.2 Two Consecutive Smallest Sectors ? 1.4.3 Four Equal Sectors ? . . . . . . . . 1.4.4 Constructing Degree4 Vertices ? . 1.4.5 HalfPlane Properties ? . . . . . . . Multivertex FlatFoldability ?? . . . . . . . 1.5.1 Isometry Conditions and Semifoldability ?? . . . . . . . . . 1.5.2 Injectivity Conditions and NonTwist Relation ?? . . . . . . . . . . . . . 1.5.3 Local FlatFoldability Graph ?? . . xv xv xviii 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 12 16 17 21 23 25 27 29 30 32 33 34 35 37 37 41 42 . . 42 . . . . 46 46 vii 1.6 1.7 Vector Formulations of Vertices ? ? ? . 1.6.1 Vector Notation: Points ? ? ? . . 1.6.2 Vector Notation: Lines ? ? ? . . 1.6.3 Translation ? ? ? . . . . . . . . 1.6.4 Rotation ? ? ? . . . . . . . . . . 1.6.5 Reflection ? ? ? . . . . . . . . . 1.6.6 Line Intersection ? ? ? . . . . . 1.6.7 2D Developability ? ? ? . . . . 1.6.8 2D FlatFoldability ? ? ? . . . . 1.6.9 Analytic versus Numerical ? ? ? Terms ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Periodicity 2.1 2.2 2.3 2.4 2.5 79 Repeating Vertices ? . . . . . . . . 1D Periodicity ? . . . . . . . . . . . 2.2.1 Periodicity and Symmetry ? 2.2.2 Tiles ? . . . . . . . . . . . . 2.2.3 Linear Chains ? . . . . . . . 2D Periodicity ? . . . . . . . . . . . 2.3.1 Huffman Grid ? . . . . . . . 2.3.2 Yoshimura Pattern ? . . . . 2.3.3 Miuraori ? . . . . . . . . . 2.3.4 Miuraori Variations ? . . . 2.3.5 Barreto’s Mars ? . . . . . . 2.3.6 Generalized Mars ? . . . . . Partial Periodicity ?, ??, ? ? ? . . . 2.4.1 YoshimuraMiura Hybrids ? 2.4.2 Semigeneralized Miuraori ? 2.4.3 Predistortion ?? . . . . . . 2.4.4 TachiMiura Mechanisms ? . 2.4.5 Triangulated Cylinders ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Triangulated Cylinder Geometry ? ? ? 2.4.7 Waterbomb Tessellation ? . . . . . . 2.4.8 Troublewit and Pleats ? . . . . . . . . 2.4.9 Corrugations and More ? . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Simple Twists 3.1 3.2 viii ........CONTENTS TwistBased Tessellations ? . . . . . . . . Folding a Twist ? . . . . . . . . . . . . . 3.2.1 Diagrams versus Crease Patterns ? 3.2.2 A Square Twist Tessellation ? . . 53 54 55 57 58 60 61 63 67 70 73 79 80 80 85 88 90 93 101 106 114 117 121 126 126 128 138 144 152 160 164 174 184 190 193 . . . . . . . . . . . . 193 195 202 206 3.3 3.4 3.5 3.6 3.7 3.8 Elements of a Twist ? . . . . . . . . . . . . . . Regular Polygonal Twists ?, ?? . . . . . . . . . 3.4.1 Cyclic Regular Twists ? . . . . . . . . . 3.4.2 Open and ClosedBack Twists ? . . . . 3.4.3 Rotation Angle of the Central Polygon ? 3.4.4 IsoArea Twists ?? . . . . . . . . . . . Twist FlatFoldability ? . . . . . . . . . . . . . 3.5.1 Local FlatFoldability ? . . . . . . . . 3.5.2 Pleat Crease Parity ? . . . . . . . . . . 3.5.3 Pleat Assignments ? . . . . . . . . . . 3.5.4 mm/vv Condition ? . . . . . . . . . . . 3.5.5 mv/vm Condition ? . . . . . . . . . . . 3.5.6 MM/VV Condition ? . . . . . . . . . . 3.5.7 MV/VM Condition ? . . . . . . . . . . 3.5.8 Cyclic Overlap Conditions ? . . . . . . 3.5.9 Summary of Limits ? . . . . . . . . . . General Polygonal Twists ??, ? ? ? . . . . . . . 3.6.1 Triangle Twists ?? . . . . . . . . . . . 3.6.2 HigherOrder Irregular Twists ?? . . . 3.6.3 Cyclic Overlaps in Irregular Twists ?? . 3.6.4 ClosedBack Irregular Twists ?? . . . . 3.6.5 OpenBack Brocard Polygon Twists ? ? ? Joining Twists ? . . . . . . . . . . . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . . . 4 Twist Tiles 4.1 4.2 4.3 4.4 4.5 Introduction to Twist Tiles ? . . . 4.1.1 What is a Tile? ? . . . . . 4.1.2 Ways of Mating ? . . . . . 4.1.3 Centered Twist Tiles ? . . 4.1.4 Offset Twist Tiles ? . . . . Vertex Figures ? . . . . . . . . . . Vertices and Angles ? ? ? . . . . . 4.3.1 Unit Polygons ? ? ? . . . . 4.3.2 Centered Twist Tiles ? ? ? 4.3.3 Offset Twist Tiles ? ? ? . . Folded Form Tiles ?, ?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Centered Twist Folded Form Tiles ?? 4.4.2 Offset Twist Folded Form Tiles ? . . . Triangle Tiles ?? . . . . . . . . . . . . . . . 4.5.1 Centered Twist Triangle Tiles ?? . . 4.5.2 Offset Twist Triangle Tiles ?? . . . . 208 211 212 214 215 216 222 225 227 228 229 230 231 232 234 238 242 243 247 249 255 262 264 268 271 . . . . . . . . . . . . . . . . 271 271 277 280 287 290 297 298 299 304 306 306 311 312 312 316 CONTENTS ........ ix 4.6 4.7 HigherOrder Polygon Tiles ?, ??, ? ? ? . . . 4.6.1 Centered Twist Cyclic Polygon Tiles ? 4.6.2 Cyclic Polygon Construction ? ? ? . . 4.6.3 Quadrilateral Offset Twist Polygon Tiles ?? . . . . . . . . . . . . . . . . 4.6.4 Offset Twist HigherOrder Polygon Tiles ?? . . . . . . . . . . . . . . . . 4.6.5 Pathological Twist Tiles ? . . . . . . 4.6.6 SplitTwist Quadrilateral Tiles ? . . . Terms ? . . . . . . . . . . . . . . . . . . . . . . . 319 319 321 . 326 . . . . 330 332 334 342 5 Tilings 5.1 5.2 5.3 5.4 5.5 5.6 Introduction to Tilings ? . . . . . . . . . . . Archimedean Tilings ?, ? ? ? . . . . . . . . . 5.2.1 Uniform Tilings ? . . . . . . . . . . . 5.2.2 Constructing Archimedean Tilings ? . 5.2.3 Lattice Patches and Vectors ? ? ? . . . EdgeOriented Tilings ? . . . . . . . . . . . 5.3.1 Centered Twist Tiles ? . . . . . . . . 5.3.2 Offset Twist Tiles ? . . . . . . . . . . kUniform Tilings ? . . . . . . . . . . . . . . 5.4.1 2Uniform Tilings ? . . . . . . . . . 5.4.2 TwoColorable 2Uniform Tilings ? . 5.4.3 HigherOrder Uniform Tilings ? . . . 5.4.4 Periodic Tilings with Other Shapes ? 5.4.5 Grid Tessellations ? . . . . . . . . . . NonPeriodic Tilings ?, ? ? ? . . . . . . . . . 5.5.1 Goldberg Tiling ? . . . . . . . . . . . 5.5.2 SelfSimilar Tilings ? ? ? . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . . 6 PrimalDual Tessellations 6.1 6.2 6.3 x ........CONTENTS 345 . . . . . . . . . . . . . . . . . . Shrink and Rotate ? . . . . . . . . . . . . . . . Properties ?? . . . . . . . . . . . . . . . . . . 6.2.1 Twist and Aspect Ratio ?? . . . . . . . 6.2.2 Crease Pattern/Folded Form Duality ?? Nonregular Polygons ?? . . . . . . . . . . . . 6.3.1 A Broken Tessellation ?? . . . . . . . 6.3.2 Dual Graphs and Interior Duals ?? . . 6.3.3 A Valid Rhombus Tessellation ?? . . . 6.3.4 Relation Between Primal and Dual Graphs ?? . . . . . . . . . . . . . . . 345 346 346 348 352 356 356 366 371 371 375 376 381 389 391 393 395 402 405 405 407 407 411 414 414 416 418 422 6.4 6.5 6.6 6.7 Maxwell’s Reciprocal Figures ?, ?? . . . . . 6.4.1 Indeterminateness and Impossibility ? 6.4.2 Positive and Negative Edge Lengths ? 6.4.3 Crease Assignment ?? . . . . . . . . 6.4.4 Triangle Graphs ?? . . . . . . . . . . 6.4.5 Voronoi and Delaunay ?? . . . . . . Flagstone Tessellations ? . . . . . . . . . . . 6.5.1 Spiderwebs Revisited ? . . . . . . . . 6.5.2 The Flagstone Geometry ? . . . . . . 6.5.3 Flagstone Vertex Construction ? . . . 6.5.4 Examples ? . . . . . . . . . . . . . . Woven Tessellations ?, ? ? ? . . . . . . . . . 6.6.1 Woven Concepts ? . . . . . . . . . . 6.6.2 Simple Woven Patterns ? . . . . . . . 6.6.3 Woven Algorithm ? ? ? . . . . . . . . 6.6.4 Flat Unfoldability ? . . . . . . . . . . 6.6.5 Woven Algorithm, Continued ? ? ? . 6.6.6 Woven Examples ? . . . . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . . 7 Rigid Foldability 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 The Easy Way or the Hard Way ? . . . . . . . HalfOpen Vertices ?? . . . . . . . . . . . . Spherical Geometry ?? . . . . . . . . . . . . A Degree4 Vertex in Spherical Geometry ?? 7.4.1 Opposite Fold Angles ?? . . . . . . . 7.4.2 Adjacent Fold Angles ?? . . . . . . . Conditions on Rigid Foldability ?? . . . . . 7.5.1 The Weighted Fold Angle Graph ?? . 7.5.2 Distinctness of Fold Angle ?? . . . . 7.5.3 Matching Fold Angle ?? . . . . . . . General Twists ?? . . . . . . . . . . . . . . 7.6.1 Triangle Twists ?? . . . . . . . . . . 7.6.2 Mechanical Advantage ?? . . . . . . NonTwist Folds ?? . . . . . . . . . . . . . . 7.7.1 General Meshes ?? . . . . . . . . . . 7.7.2 Quadrilateral Meshes ?? . . . . . . . NonQuadrilateral Meshes ? . . . . . . . . . 7.8.1 Forced Rigid Foldability ? . . . . . . 7.8.2 NonFlatFoldable Vertices ? . . . . . Terms ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 425 429 433 435 438 442 443 446 447 450 453 454 456 459 461 465 467 471 475 475 477 480 486 486 489 492 495 498 500 506 508 512 515 515 518 528 528 530 533 CONTENTS ........ xi 8 Spherical Vertices 8.1 8.2 8.3 8.4 8.5 8.6 Generalizing Vertices ? . . . . The Gaussian Sphere ?? . . . 8.2.1 Plane Orientation ?? . 8.2.2 The Trace ?? . . . . . 8.2.3 Polyhedral Vertices ?? 8.2.4 A Degree4 Vertex ?? Sector and Fold Angles ?? . . 8.3.1 Osculating Plane ?? . 8.3.2 Binding Condition ?? 8.3.3 Ruling Plane ?? . . . 535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?? 8.3.4 Real Space Solid Angle . . . . . . . 8.3.5 Ruling Angle ?? . . . . . . . . . . . . 8.3.6 Osculating Angle ?? . . . . . . . . . . 8.3.7 Adjacent Fold Angles ?? . . . . . . . . 8.3.8 FlatFoldable and StraightMajor/Minor Vertices ?? . . . . . . . . . . . . . . . 8.3.9 Sector Angle/Fold Angle Relations ?? . More Angles and Planes ?? . . . . . . . . . . 8.4.1 Sector Elevation Angles ?? . . . . . . 8.4.2 Sector Angles ?? . . . . . . . . . . . . 8.4.3 Bend Angle ?? . . . . . . . . . . . . . 8.4.4 Edge Torsion Angle ?? . . . . . . . . . 8.4.5 Midfold Angles and Planes ?? . . . . . 8.4.6 Infinitesimal Trace ?? . . . . . . . . . 8.4.7 What Specifies a Vertex? ?? . . . . . . Networks of Vertices ?? . . . . . . . . . . . . 8.5.1 Huffman Grid ?? . . . . . . . . . . . . 8.5.2 Gauss Map ?? . . . . . . . . . . . . . 8.5.3 Miuraori and Mars ?? . . . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . . . 9 3D Analysis 9.1 9.2 9.3 xii ........CONTENTS 3D Vectors ? ? ? . . . . . . . . . . . . . . . 3D Vertices ? ? ? . . . . . . . . . . . . . . . 9.2.1 Fold Direction Vectors ? ? ? . . . . . 9.2.2 Vertex from Fold Directions ? ? ? . . 9.2.3 Degree4 Vertex from Sector Elevation Angles ? ? ? . . . . . . . . . . . . . Discrete Space Curve ? ? ? . . . . . . . . . . 535 536 536 538 541 543 545 545 547 549 550 554 557 558 561 563 568 569 573 576 578 584 586 590 591 591 593 597 603 605 . . . . 605 610 611 612 . . 615 617 9.4 9.5 9.6 9.7 Plate Model ? ? ? . . . . . . . . . . . . 9.4.1 Folding a Crease Pattern ? ? ? . 9.4.2 Fold Angle Consistency ? ? ? . 9.4.3 Solving for Fold Angles ? ? ? . Truss Model ? ? ? . . . . . . . . . . . . 9.5.1 3D Isometry and Planarity ? ? ? 9.5.2 Explicit Stress/Strain ? ? ? . . . 9.5.3 3D Developability ? ? ? . . . . Time Efficiency ? . . . . . . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Rotational Solids 10.1 10.2 10.3 10.4 10.5 10.6 10.7 ThreeDimensional Twists ?, ?? . . . 10.1.1 Puffy Twists ? . . . . . . . . . 10.1.2 Folding a Sphere ?? . . . . . ThinFlange Algorithm ? ? ? . . . . . ThickFlange Structures ?, ? ? ? . . . 10.3.1 Mosely’s “Bud” ? . . . . . . . 10.3.2 ThickFlange Algorithm ? ? ? 10.3.3 Specified Gores ? ? ? . . . . . 10.3.4 Generalized Flanges ? ? ? . . Axial Unfoldings ? ? ? . . . . . . . . Variations on the Theme ? ? ? . . . . 10.5.1 Twist Lateral Shifts ? ? ? . . . 10.5.2 Triangulated Gores ? ? ? . . . Artists of Revolution ? . . . . . . . . Terms ? . . . . . . . . . . . . . . . . 626 627 628 633 636 637 641 644 646 647 649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 649 653 659 664 664 668 671 674 679 683 683 695 699 702 Afterword 705 Acknowledgements 707 Bibliography Index 711 723 CONTENTS ........ xiii Introduction ? 1. Genesis Everyone and no one knows what origami, the Japanese art of paperfolding, is. Everyone knows, because they have seen the wellknown Japanese crane, or tsuru, an international symbol of peace. Or they have folded schoolyard paperfolding—boats, bangers, and cootiecatchers, possibly not even knowing that these, too, are part of the origami world. But in another sense, no one knows what origami is, because in the latter part of the 20th century, it exploded in variety, complexity, and artistry, with numerous genres and specializations. The word origami is simply the Japanese words for “folding” (oru) and “paper” (kami), combined in a single word used to describe the craft—and sometimes, art—of decorative paper folding. Contrary to popular belief, there are no fixed rules about paper, glue, or use of cuts: traditional Japanese designs, many of which can be reliably dated to be hundreds of years old, used various sizes and shapes of paper, sometimes multiple sheets, and often used cuts. In modern origami, cuts are rare but not unknown, and distinct genres have arisen in which one folds a single sheet or uses multiple sheets (the latter category divided further into composite origami, using multiple sheets to make separate parts of a subject, and modular origami, using multiple sheets to make identical units that are assembled). The most wellknown origami genres are representational and figurate; the origami subject looks like something. Indeed, most people who have ever seen or folded origami have only created representational work. But there is a deep history of nonrepresentational decorative paperfolding, both within xv the world of Japanese origami and coming from many disparate fields of endeavor outside of the Japanese tradition, ranging from napkinfolding of the 15th century in Europe to early20thcentury Bauhaus architecture to late20thcentury computational geometry and mathematics. It is this latter field of nonrepresentational origami that is the focus of this book, centered around the mathematical genre known as origami tessellations. A tessellation is, in general, a division of the plane into a pattern, and the name comes from the Latin tessera, which was the name for a tile making up a mosaic. Like their namesakes, origami tessellations divide the plane into decorative patterns—but using folds to make the subject from a single sheet of paper, rather than dividing the image up into individual units.1 Tessellations have a distinct history within origami, with independent original work by both Shuzo Fujimoto and Yoshihide Momotani in the 1970s and 1980s, but they underwent a renaissance in the late 1990s when the concept was picked up and explored by several artists, notably Paulo Taborda Barreto and Chris K. Palmer in the 1990s, and many more thereafter. Their work, in turn, inspired still further development in the world of origami and the beginnings of research into folded tessellations’ mathematical properties and algorithms for their design, launching a period of growth and exploration that continues to this day. Along with the expansion of origami tessellations has come an expansion of the scope of interest, as origami “tessellators” began exploring other folded forms that bore similarities to tessellations but were not particularly flat, periodic, or reminiscent of any sort of tiling. They weren’t necessarily flat and were often threedimensional. What these new foldedpaper forms had in common were that they were nonrepresentational and often highly geometric. The origami tessellation explorers also discovered something else: mathematical and nonrepresentational folding had a welldefined existence quite outside of the world of origami. Notably, Josef Albers and the Bauhaus school explored folded paper extensively in the 1920s, including flat, polyhedral, and curved forms. 1 In fact, there are modular origami tessellations, patterns that more closely resemble the traditional type of tessellation mosaic in which individual units are folded and then assembled into an overall pattern. I note their existence but won’t describe them further in this work. xvi ........INTRODUCTION Many of the shapes they explored have been rediscovered by modern artists quite independently. The universality of mathematics almost guarantees that a simple elegant form will be discovered over and over by multiple artists. Paper artists, too, have discovered and created geometric and mathematical forms through the years, again, quite independently of the origami tradition. Paper is quite a versatile medium: it can be cut, folded, scored, bent, dampened, reshaped, and altered in a variety of ways. It can be made to take a crease, giving it a memory and a desire to take on certain forms; yet, the springiness of paper can make it resist certain forms as well. This versatility, coupled with its generally low price, has made paper the medium of experimentation for artists and designers of all stripes. It has also been the inspiration of mathematical research into the properties of folded shapes, again, outside of the traditional world of Japanese paperfolding. Some 40 years ago, computer scientist and artist Ron Resch began designing and folding paper forms using mathematical and computational algorithms. Resch was followed by another computer scientist, David Huffman, who, beginning in the 1970s, not only built on Resch’s ideas but developed many new concepts of his own and eventually wrote one of the seminal papers on the mathematics of folded paper. Their work launched a new thread of research into geometric paperfolding, in which mathematics and art were combined in equal proportion: algorithms, existence, and complexity paired with statement, expression, and aesthetic. The world of geometric origami preceded that subset that we now call origami tessellations, and it has, in recent years, grown far beyond mere tessellations. The repeating patterns of tessellations, however, have a particular appeal to me, and, for the purposes of this book, provide a unifying theme for a study of primarily geometric, singlesheet folding. And so with this book, I have placed origami tessellations in both the title and at the heart of the book. Tessellations are beautiful, but they also provide a structured way to introduce the mathematical laws governing origami—laws that govern in so many ways what structures, forms, and shapes can be created by folding. But this book covers more than tessellations; I will range over many different geometric forms. It is not possible to be truly comprehensive, because the field of geometric origami is growing actively, in many directions at once. I hope to pull together here a sampling of many of the possibilities and to provide INTRODUCTION ........ xvii you with tools, both artistic and mathematical, that you can use to reproduce the artworks and patterns within this book and to build and fold your own original creations. ? 2. What to Expect and What You Need Mathematical origami is an extremely diverse field with many branches, only some of which are represented in this book. (It is also a fastgrowing field; by the time you read this, there will likely be many new shoots.) Thus, you can, and indeed are invited to, jump among the various chapters, trying out things that look interesting and skipping what makes your eyes glaze over or is packed with forbiddingly complex expressions. It’s no fun at all to be sampling the treats at the table and unwittingly find yourself with a mouthful of something only a connoisseur should appreciate. This metaphor applies particularly to things mathematical. The mathematics in this book covers a wide range of topics and requires a wide ranges of skills. To help you decide what to jump into and what to skim past, I have marked the subsections with from one to three stars, indicating the level of mathematics required: ? Basic. The simplest geometry, requiring little more than an appreciation of shapes, the ability to construct and/or measure an angle with a protractor, and an ability to count, add, and subtract angles. We will use some letters to represent quantities, but we will keep the algebra to a minimum. Suitable for early high school students. ?? Intermediate. Uses algebra (equationsolving), trigonometry, and more advanced concepts from high school geometry. Suitable for upper high school students. ? ? ? Advanced. Uses concepts from linear algebra, vectors and operators. Suitable for college students in technical fields (mathematics, science, engineering) and possibly some advanced high school students.2 2 In the ? sections, angles will be given in terms of degrees. In ?? and ? ? ? sections, we will use radians. xviii ........INTRODUCTION You will also find origami instructions for several figures at various places. Traditional origami instructions are given in terms of a folding sequence: a stepbystep series of drawings showing a linear progress from the unfolded paper to the fully folded result. However, more often than not, mathematical origami has no folding sequence, which creates its own set of challenges when it comes to presenting instructions for such works. And it’s not just a case of the designer being “too lazy to draw diagrams.” As we will see, many mathematical folds are composed of large irreducible blocks of folds—structures that cannot be broken down into isolated steps. Historically, most origami designs were discovered by sequential manipulations performed on a sheet of paper, and so no matter how long and/or convoluted the path to the end result, it was pretty much guaranteed that a folding sequence existed; the challenge of diagramming it lay primarily in remembering (or reconstructing) the most efficient sequence, leaving out all of the exploratory dead ends. But with the modern age of “technical origami,” or origami sekkei, in which the final form is designed before one ever puts hand to paper, there is no reason to believe that a simple path from start to end exists—and, in many cases, it is possible to show mathematically that such does not exist. In this regard, mathematical origami shares a property with what in some ways is its exact opposite: highly sculpted representational origami, most famously typified by the works of the late Eric Joisel. Joisel called his work “jazz origami,” because the vast majority of the folds were improvised on the spot based on aesthetic considerations. Here, too, there is no set sequence, no set of diagrams that can provide instruction; instead, Joisel simply moved around the paper, bending, shaping, curving, adding folds, nudging it ever closer to the ideal he visualized, but in no set order. Surprisingly, many mathematical folds require a very similar approach: since tens or hundreds of creases may need to come together at once, the artist must simply work his or her way around the crease pattern, the design, bending each fold in the proper direction but in no particular order, until they all (or a large subset) can come together. One nice property that many mathematical folds have is a “tipping point”—a point at which the number of creases going in the right direction reaches a critical mass and the fold, instead of resisting, starts to come together, almost with a life of its own. INTRODUCTION ........ xix So mathematical origami breaks new ground in its design and in how it goes together, which is to say, even if you don’t have much past experience with origami, that is not much of a handicap. The field of mathematical origami is so new that, in some sense, none of us is very far from the beginning. It is that beginning to which we now turn. xx ........INTRODUCTION 1 Vertices ? 1.1. Modeling Origami The term origami refers to something very specific: Japanese paperfolding. But mathematical origami is much broader than the traditional craft: it isn’t necessarily Japanese, it involves materials other than paper, and it involves actions other than just folding— bending and crumpling, for example, although both could be said to be a form of folding. What we will focus on in this book, though, are those aspects of mathematical folding that are characteristic of most origami: the use of a nonstretchy sheetlike material, manipulated in three dimensions, with few or no cuts. Mathematical folding doesn’t require that you use paper—in fact, in realworld applications of mathematical folding, one can use materials as diverse as plastic, Mylar, Kapton, leather, cloth, and even mats of carbon nanotubes. But throughout this work, for simplicity of language, I will generally refer to the material being folded as “paper,” and paper is often the ideal material to work with: inexpensive, widely available in diverse forms, and possessed of mechanical properties that make it particularly suited for folding. Part of the beauty of origami in general and mathematical folding in particular is that it is tactile and visual; you can feel the paper, you can see the result, and integration of handeye experience builds an intuition of what is possible more effectively than any set of mathematical formulas or algebraic description. Nevertheless, there are limits to intuition, and mathematics can provide powerful tools to understand the possibilities of paper and to design specific structures and forms. And so, throughout this work, I will attempt to provide a mathematical description of the topic at hand. 1 There are many ways to describe folding mathematically, and the most natural way depends in large part on the level of abstraction that one chooses in the description. Is the folded form flat or threedimensional (3D)? Are surfaces straight or curved? Are creases straight or curved? Do we care about effects of material thickness, tensile forces, mechanical yield, creep, and plastic deformation? There is no single “correct” mathematical description of folded paper; there are only various approximations that idealize, emphasize, and/or ignore different aspects of the folding process. Two properties stand out above others as necessary to describe what is recognized as origami and that play a role in nearly all mathematical descriptions: • Nonstretchy paper. The folded shape is a 3D deformation of a planar surface that does not appreciably stretch (or compress) in any direction. • Nonselfintersection. The paper cannot intersect itself in the folded form, or in any intermediate stage. Any mathematical description of paperfolding must include these two properties in some way or another. These two properties—nonstretchiness and nonselfintersection—are at the heart of the folding arts. It is a little awkward to describe the properties of paper by what it is not; better to have a positive term. There are terms for both nonstretchiness and nonselfintersection. When we say that the paper is not stretchy, we mean that if we draw a line on the paper, fold the paper, and then measure the length of the line along the paper, that is, following the path of the paper, the length is unchanged. This property is a quality called isometry—taken from the Greek iso, meaning “same,” and metry, meaning “measurement.” So the essence of origami folding is that it is isometric: distances along the surface of the paper are preserved going from the flat to the folded state (and, ideally, in all intermediate states). The second property, that the paper cannot intersect itself, also has a mathematical name: injectivity. In the language of mathematics, a mapping from one set (the domain) to another (the range) is an injection if no two points in the domain map to the same point in the range. In real physical origami, we cannot have 2 ........CHAPTER 1. VERTICES two points on the paper occupy the exact same point in space when the paper is folded. Even if you fold two layers together, one layer must lie above or below the other. If two layers switch places— here layer 1 lies above layer 2, there layer 2 lies on top—the rearrangement must happen without the paper penetrating itself, neither in unfolded layers nor at a fold. So injectivity is the quality of nonselfintersection. These two qualities are what define the mathematics that are particular to origami. This is not to say, however, that every mathematical model of origami must strictly have these two qualities. In fact, as we will see, it is frequently convenient to model origami paper as a zerothickness surface, in which case a stack of layers may very well violate injectivity by occupying the exact same position in mathematical space. The important thing in such cases, though, is that in such a model, we know that the mathematical idealization violates one or the other of the fundamental properties of origami. Frequently, we will patch up such an ideal mathematical model to recover the lost properties. A mathematical description of origami must also make some assumption about the folding process, that is, the way that the paper gets from its initial flat state to its final configuration, the folded form. In standard origami books, that process is a relatively linear sequence of small steps: fold the paper in half; unfold; squashfold; petalfold; and so forth, where each term (“squashfold,” “petalfold,” etc.) refers to a specific manipulation involving a small number of folds at a time. While this linear stepbystep process was historically the most common form of origami, it is not the only way a folded figure can take form. In fact, as we will see, many of the creations of mathematical folding come together only with tens, or even hundreds, of creases moving at once. When we take the process of formation into account, mathematical descriptions and modeling can get very complex indeed; there are folds that “don’t exist,” meaning that within some mathematical system, the motion going from a valid form at step A to a valid form at step B takes an intermediate state that somehow violates the assumptions of the mathematical system and thus, according to the mathematical model, could not be folded so. There are also folds where the folded state exists (within the mathematical system), and the unfolded state exists (within that same system), but there is no smooth progression from the unfolded state to the folded state within the same mathematical sysCHAPTER 1. VERTICES ........ 3 Model FlatFoldable Origami Polyhedral Origami Curved Origami Thick Origami Description All facets are flat and coplanar; creases have fold angle of 0◦ or ±180◦ ; paper has zero thickness. Facets are flat, creases are straight, but fold angles can vary continuously; paper has zero thickness Facets and creases can be curved; paper has zero thickness Paper thickness is explicitly included. Table 1.1. Hierarchy of mathematical models of origami. tem. An example might be the mathematical model in which all surfaces apart from the folds are flat and planar. It may be that the only way to actually fold the paper into the finished state is to curve and/or bend some regions of the paper. If our mathematical system does not allow curving or bending, then we would say that, within that mathematical system, the folded state is “impossible” to fold. Whether a folded state or folding process is “impossible” or not depends, of course, on the mathematical model that one uses to describe it. If we can fold the object in the real world, then surely it exists, whatever the mathematical model might say! We must, of course, always realize that a mathematical model of folding is at best an approximation of what really happens in the physical world. The value of such a model, even as an imperfect approximation, comes when it can provide a reasonably accurate prediction of the folded state, and usually, the simpler the model, the better. We can construct something of a hierarchy of origami modeling of increasing complexity as we relax the rules of folding, as shown in Table 1.1. In general, as one moves down this hierarchy, the mathematical complexity increases—sometimes dramatically. We will explore this hierarchy, but we will move through it gradually, building base camps along the way and scheduling copious rest days as needed. And we will begin with the simplest possible model, which, surprisingly, covers a great deal of both historical and modern paperfolding. The first description we will consider is what for many years was the most common description within mathematical origami, and it is very simple indeed. In this description, we make these simplifying assumptions: 4 ........CHAPTER 1. VERTICES 1. The paper has zero thickness. 2. The folded form is flat. 3. We don’t care about any intermediate configuration, i.e., whether it is flat or 3D or theoretically possible within our model. We call this model of origami flatfoldable origami. Such a model is, of course, an approximation of reality; there is no such thing as zerothickness paper, and there is no way that an unfolded crease pattern can discontinuously transform itself into a folded state. Indeed, it is possible to contemplate folded configurations for which there is no practically achievable folding sequence. Nevertheless, this simple model can accurately describe a great deal of historic and modern folding, and it contains surprising richness and depth. This model can provide practical recipes and algorithms for the construction of folded shapes that are beautiful, interesting, and practically useful. ? 1.1.1. Crease Patterns A feature of this simplest type of origami, what we call flatfoldable origami, is that in the folded form, all surfaces are flat, except along straight lines, which are the creases, and the creases meet in groups at points, called vertices. The flat regions bounded by the creases are facets. There is a onetoone mapping between points in the original paper and points in the folded form, and we can identify each point in the original paper as to whether it ends up in a facet, a crease, or a vertex. Logically enough, we call the points facet points, crease points, or vertex points, respectively. We can then, if we like, decorate the paper with identifying information, coloring each point and line according to its status in the folded form. Such a decoration is called the crease pattern associated with the folded form. The crease pattern is, essentially, a minimal description of the origami figure. For flat origami, often the crease pattern alone suffices as a guide for how to fold the shape. The crease pattern has a long history within origami; Figure 1.1 shows an origami crease pattern (and folded form) from 1845 (reprinted in [13, p. 58]). In historical origami works and works of the early 20th century, crease patterns were not uncommon (see, e.g., [130, pp. 24–26]), but with the growth of stepbystep instructions, they began to fall ........ CHAPTER 1. VERTICES 5 Figure 1.1. Crease pattern and folding instructions for “Ono no komachi” (a female poet), from the Kayaragusa, a collection of paperfolding instructions from 1845. 6 ........CHAPTER 1. VERTICES Figure 1.2. Top: left and middle, the fold and result for a valley fold; right, the crease pattern for a valley fold. Bottom: left and middle, the fold and result for a mountain fold; right, the crease pattern for a mountain fold. out of favor. With the resurgence of mathematical folding and systematic design toward the end of the 20th century [68], though, crease patterns have returned as the blueprint of all of the folding that is to follow, and they will be a key concept throughout this book. In representational folding, the crease pattern (often referred to simply as the CP) is rarely a map of all of the folds in the design; usually, it is a selected subset, chosen by the artist to convey the important properties of the structure and/or internal symmetries. In geometric folding, by contrast, the CP is quite often comprehensive, containing every crease in the finished work. Even so, it often does not provide a full description of the origami figure. It may contain all the folds, but it says nothing about the order in which the folds are made. And many crease patterns, including the one in Figure 1.1, don’t even tell which direction the paper folds. In a flat origami figure, every fold can go in one of two directions, as shown in Figure 1.2. In conventional origami terminology, when you fold a flap toward you, the resulting fold is called a valley fold. When the flap is folded away from you, the resulting fold is called a mountain fold. Historically, valley and mountain folds were not distinguished in any way (as in Figure 1.1), but in the mid20th century, Akira ........ CHAPTER 1. VERTICES 7 Yoshizawa in Japan and Robert Harbin and Samuel L. Randlett in the West adopted a standard for diagrammatic origami instruction in which valley folds were indicated by a dashed line and mountain folds were indicated by a chain line (dotdotdash). These conventions are now widely established in stepbystep origami instructions and have become the international language of origami instruction. In a crease pattern, every fold line can be specified as to whether it is a valley fold or mountain fold in the folded form. This specification is called a crease assignment (or just assignment) of the crease pattern. It would seem natural to use the standard dashed and chain lines that are used in origami diagrams, but for crease patterns, they don’t work as well as they do in stepbystep instructions. Dashed lines and chain lines stand out when there are only a few of them, but for complex crease patterns, which arise in both figurate and geometric origami, they dissolve into a visual morass of indistinguishable strokes. For crease patterns, which can contain hundreds of folds, we need to adopt drawing conventions that provide a much stronger visual distinction between mountain and valley lines. In contrast to stepbystep origami diagrams, there is no standard convention yet for crease patterns—in part because the many variables of line pattern, thickness, hue, and saturation can be used to convey a wide range of information beyond simple valley or mountain status. No single attribute is ideal: varying the line weights degrades when a pattern is photocopied; color also doesn’t copy well (and, depending on choice of color, can fail for colorblind readers). If one does use dash patterns, they need to be strongly contrasting, even when viewed at a distance. The most robust convention would be to use all available attributes: line weight, color, saturation, and dashing. There is starting to be a consensus that in complex crease patterns, mountain folds should be dark, less saturated, and solid, while valley folds should be lighter, possibly more saturated in color, and, ideally, dashed, for the colorblind or otherwise visually impaired.1 1 Why are mountains the creases that are dark and solid in crease patterns? Generally, a mountain fold crease—the crease obtained by unfolding a mountain fold—exhibits greater contrast with the surrounding paper than a valley fold crease, so we give mountain folds the line style with stronger contrast in crease patterns. 8 ........CHAPTER 1. VERTICES Figure 1.3. Left: a crease pattern using conventional mountainvalley line patterns. Right: the crease pattern using CP coloring. The conventions I will use throughout this book are shown in Figure 1.2 on the right. I call this scheme CP coloring. The difference in visual perception and comprehension between the old and more recent representation systems can be striking; Figure 1.3 illustrates the same crease pattern with the two different drawing conventions. I will use this convention throughout for crease patterns. For stepbystep diagrams, however, I will continue to use the conventional dashed (valley) and chain (mountain) lines. The crease pattern can serve as a plan for the folded figure (though that is not its only role). Even as a plan, though, it is not a complete plan, in the sense of providing a complete description of the folded form. Not only does it fail to specify the temporal order in which one might form the creases, it doesn’t necessarily fully specify the stacking order of the facets in the folded form. One could, of course, just make up a stacking order for the facets, but if we choose a stacking order, that will imply a particular crease assignment. It might also imply that the paper intersects itself—which, in the real world, is not allowed. If a crease pattern can be folded with physical paper, i.e., with no stretching or selfintersection, then it is a valid crease pattern. Similarly, a stacking ........ CHAPTER 1. VERTICES 9 Figure 1.4. Left: a crease pattern of two valley folds. Middle, right: two different stacking orders of the facets in the folded form. A B C A B C Figure 1.5. Left: a crease pattern of two valley folds. Right: one of the two stacking orders in the folded form. order on a crease pattern is valid if and only if it does not imply any selfintersection. Even with a complete crease pattern, determination of a valid stacking order can be computationally extremely challenging, even intractable. While a fuller analysis of this point requires analysis from the world of computational complexity [11], I would like to point out a very simple example that hints at the potential difficulties. Consider, for example, the crease pattern shown in Figure 1.4, consisting simply of two valley folds. There are two possible stacking orders for the facets, even though the creases are exactly the same. In this pattern, the two possibilities are rather obvious, but in complex crease patterns, there can be subtle and longrange interactions between parts of the crease pattern that limit potential stacking orders. Consider, for example, the crease pattern in Figure 1.5, similar to the preceding, but in which the two vertical valley folds divide the strip evenly into thirds. This pattern, too, admits two stacking orders, in which either facet A or facet C can wind up on top. But if facet A is just the tiniest bit wider than facet B, one of the two stacking orders is no longer possible; facet C can’t wind up on top because that would force facet A to penetrate the right crease. Similarly, if facet C were just a bit 10 ........CHAPTER 1. VERTICES wider, that would ensure that facet A could not wind up on top because that would force facet C to penetrate the left crease. And if both facets A and C were wider than B, this would be an invalid crease assignment: it would not be possible to make both folds at the same time in the specified direction. This relationship should set off some warning bells. What happens at the crease between facets B and C on the right depends critically on details at the far left of the crease pattern—namely, how far to the left facet A extends. The same logic applies to the crease between facets A and B. In general, every crease pattern has the potential for such nonlocal interactions between its constituent parts. The foldability of the pattern, and/or the validity of its crease assignment, can depend on relationships between farflung features of its crease pattern. We will encounter many such examples in our explorations of tessellations and other mathematical folds. One more note on my schematic representations of origami forms: it is customary in origami to fold from paper that is colored on one side and white (or contrastingly colored) on the other. Although this practice is by no means necessary, it is often helpful to distinguish between the two sides of the paper. So, as I have done in Figures 1.4 and 1.5, I will usually show the two sides of the paper in contrasting colors and will refer, where appropriate, to the “white side” and “colored side” of the paper. Crease patterns will usually be drawn on the white side, by convention, and also, for better legibility and contrast. Although it is an incomplete description of a fold, a crease pattern is a very useful tool for concisely describing the structure of a folded shape, and while in principle the stacking order may be difficult to discern from the crease pattern, for the vast majority of folds of practical interest, the preferred stacking order is readily found. Thus we begin a long and fruitful relationship with crease patterns in this book. I provide many crease patterns as illustrations throughout, and I encourage you to reproduce them and try folding them up as you work your way through the book. Not just any pattern of lines can serve as an origami crease pattern. In fact, there are several highly restrictive conditions that apply that determine whether a crease pattern can be folded at all, whether it keeps the facets flat or forces them to bend, and whether it allows the paper to be entirely flattened. Such conditions are important: they tell us what is possible and impossible, ........ CHAPTER 1. VERTICES 11 and from among the possible, they provide guidance to accomplishing desired objectives—design rules, in other words. We start our journey with the simplest, most ideal form of origami: flatfoldable origami and their crease patterns. And we will start our study of such crease patterns with the building block of crease patterns: the crease. ? 1.1.2. Creases and Folds Crease patterns are made up of two types of geometric objects: points and lines. The points, or vertices, are places where lines come together. The lines, of course, are the creases themselves. We can identify a higher level of structure in crease patterns: crease lines outline facets. In the same way that crease lines are bounded by vertices (one at each end), facets are bounded by crease lines, by the border of the paper, or a combination thereof. As we have seen, there are constraints on crease patterns: not all crease assignments give valid, i.e., physically foldable, forms. There are also constraints on the angles of the creases relative to one another, which determine whether the origami figure can, in fact, truly fold flat. And there are constraints on the stacking order of the facets. All of the constraints arise in order to satisfy isometry and injectivity (from Section 1.1). In flat origami crease patterns, a crease line can take on one of three states: it can be a valley crease, a mountain crease, or an unfolded crease, i.e., one that is flat. Figure 1.2 showed a valley and a mountain fold. The third possibility is, simply, no fold at all. Figure 1.2 illustrates the difference between a valley and a mountain fold. If you are looking at the white side of the paper, a valley fold brings the moving part of the paper toward you, while a mountain fold moves it away from you. But we can also define valley and mountain folds in terms of the fold angle. By convention, the fold angle is defined as the deviation from flatness of the intersection between the paper and a plane perpendicular to the fold, as illustrated in Figure 1.6 on the left. By this convention, an unfolded crease has a fold angle of 0◦ , which fits with the concept of “not folded.” Similarly, mountain folds have the same magnitude of fold angle as valley folds, but opposite sign. Swapping the parity of folds—changing all mountain folds to valley folds and vice versa—is the same as changing the sign of all folds. 12 ........CHAPTER 1. VERTICES g >0 +180¡ 0¡ +180¡ g =0 0¡ +360¡ g <0 180¡ Figure 1.6. Definition of the fold angle γ. Left: a flatfolded crease has a fold angle of (top to bottom) +180◦ for a valley fold, 0◦ for an unfolded fold, and −180◦ for a mountain fold. Right: a flatfolded crease has a dihedral angle of 0◦ for a valley fold, +180◦ for an unfolded fold, and +360◦ for a mountain fold. A closely related measure of angle is the dihedral angle, which is typically taken to be the angle measured between two facets, as illustrated in Figure 1.6 on the right. The fold angle and dihedral angle are simply related: fold angle = 180◦ − dihedral angle. In origami analysis, the fold angle is usually the more natural way to characterize angles: • An unfolded crease has a dihedral angle of 180◦ and a fold angle of 0◦ , with the latter value corresponding to the idea of “no fold.” • Mountain and valley folds have dihedral angles of 360◦ and 0◦ , respectively, and fold angles of −180◦ and +180◦ , respectively, with the latter capturing the idea that a mountain fold is the opposite of a valley fold. ........ CHAPTER 1. VERTICES 13 Normally, a crease is identified as a mountain or valley fold based on the perspective of the viewer: a fold is a valley fold if it folds toward the viewer, whether or not the viewer is looking at the white or colored side of the paper. I will usually draw crease patterns as viewed from the white side of the paper, to provide greater contrast and visibility for the crease lines. In a flat origami crease pattern, all creases are one of mountain, valley, or unfolded. For convenience, we will often label these M, V, or U, respectively. A flat origami crease pattern whose lines have been labeled with their fold angle by color and/or line pattern is said to be creaseassigned (or just assigned, for short). One might wonder why one would include unfolded (U) creases at all; if the paper is unfolded everywhere within the facets, then what’s the distinction between an unfolded crease and no crease? It turns out that in the world of origami design, it is not uncommon to construct a crease pattern in two phases: first, compute the locations of all possible creases; second, assign those creases to be mountain, valley, or unfolded, depending on factors that relate to layer ordering, flap position, and the like. So we will consider the possibility of unfolded creases. If a crease pattern consists of only mountain and valley creases (no unfolded creases), we will call it a fully folded crease pattern. Many of the laws of crease assignment, such as the MaekawaJustin Theorem, apply only to fully folded patterns. When this is the case will usually be clear from context; if the situation is ambiguous, I will make it explicit. We will often consider crease patterns in which the crease lines have not been assigned; such a pattern is called an unassigned pattern, naturally enough. In unassigned patterns, all crease lines will be drawn in the same way, as unfolded crease lines, as shown in Figure 1.7 (and sometimes we will show them as heavier lines, if greater contrast is desired). We also point out a property visible in the pattern in Figure 1.7 that is a universal property of flat origami crease patterns: if two facets are incident on a common folded crease in the crease pattern, then in the folded form, one of the facets must be whiteup (white side facing the viewer) and the other facet must be colorup (colored side facing the viewer). Two facets incident to a common unfolded crease must, of course, have the same orientation in the folded form. 14 ........CHAPTER 1. VERTICES Figure 1.7. Top: an unassigned crease pattern. Middle: the assigned crease pattern. Bottom: the folded form corresponding to this assignment. Often when one is developing an algorithm related to the analysis of crease patterns, it is useful to consider the border of the paper to be a (special) type of crease: a border crease. This assumption means that every facet is bounded by some type of crease: mountain, valley, unfolded, or border. (Since border creases are incident to only a single facet, the fold angle for a border crease would be undefined.) We could, if we wished, mark each facet of the crease pattern in such a way as to indicate which way it faces in the folded form, whether it is white side up or colored side up. An example of this coloring for a simple origami model (Sam Randlett’s “New Flapping Bird” [102, p. 126]) is shown in Figure 1.8. Here we are looking at the white side of the crease pattern, but I have given a slightly darker tint to those facets that end up colored side up in the folded form. Whenever we cross a fold in the crease pattern, we must be moving from a whiteup facet to a colorup facet or vice versa. Thus, for any flat origami crease pattern, if we color the facets according to whether they are whiteup or colorup, this coloring has the property that no two facets of the same color meet along a common fold line. This is called a twocoloring of the crease pattern. Every flat origami crease pattern can be twocolored (remember, you only include crease lines that are actually folded); ........ CHAPTER 1. VERTICES 15 Figure 1.8. Left: crease pattern; darker facets are colored side up in the folded form. Right: folded form. the marking of which facets are whiteup or colorup provides such a coloring. For any given twocolorable pattern, there are only two possible twocolorings: one is just the reverse of the other. Thus, we can go the opposite direction as well; given a flat origami crease pattern, each of the two possible twocolorings of that pattern automatically gives a map of which facets are whiteup and which are colorup in the folded form—this without having to actually fold the pattern up, or even know which folds are mountain or valley. ? 1.2. Vertices Within a crease pattern, crease lines come together at points called vertices, and it is there that the conditions of flatfoldability begin to apply. First, we should define a bit of terminology. We have already talked about fold angles, the angles made between the facets on either side of a crease. At vertices within a crease pattern, we are concerned about the angles between the crease lines themselves. We call these the sector angles at the vertex. A hypothetical vertex is shown in Figure 1.9; each of the sector angles is labeled with the Greek letter theta (θ), subscripted by i, where i is the index of the sector angle. By convention, we will both number and index angles going counterclockwise (CCW) around any vertex, as we have done here. That is, we number the fold lines 1, 2, . . .. The ith sector angle is then the angle between the ith crease line and the next crease line going around the vertex. 16 ........CHAPTER 1. VERTICES f3 q3 f4 q4 f2 q2 q1 f1 Figure 1.9. É Schematic of a vertex with labeled sector angles {θ i } and fold direction angles {φi }. f5 It is also possible to characterize each crease line by its fold direction angle, i.e., its angle measured with respect to some reference, typically an imaginary horizontal line emanating to the right. We will denote this fold direction angle by φi for the ith crease line (using the Greek letter phi). In Figure 1.9, the fold direction angle of the first crease line, φ1 , is 0◦ , since this crease line runs horizontally. The fold direction angles of all other crease lines are measured with respect to this reference. The sector angles are simply the difference in angle between two consecutive crease lines, and so for most of the sectors, we have that θi = φi+1 − φi . (1.1) This is except, of course, for the last sector angle, when the fold direction angles wrap around from 360◦ to 0◦ . We can handle this case by modifying the definition of sector angle: θi = (φi+1 − φi )mod 360◦, (1.2) where “mod 360◦ ” means that we add or subtract multiples of 360◦ to the value until it lies within the range [0, 360◦ ). (Incidentally, that’s not a typo. In mathematics, using a square bracket means the range includes the endpoint, while using a parenthesis means it doesn’t, so [0, 360◦ ) means that 0 is included in the range but 360◦ isn’t.) Now that we know how to talk about angles around a vertex, we are ready to say something about those angles. ? 1.2.1. KawasakiJustin Theorem The first property of vertices in a crease pattern ultimately derives from the condition of nonstretchiness (isometry), which manifests ........ CHAPTER 1. VERTICES 17 Figure 1.10. Left: we cut off a circular region around a vertex in the folded form. Middle: the circlecut corner. Right: the crease pattern of the vertex after unfolding. itself at a vertex in the property that all sector angles are unchanged in moving from the crease pattern to the folded form. Let us consider a small thought experiment. Suppose that we have a complex folded origami figure (see Figure 1.10); we identify a single vertex, cut off a small circular arc around that vertex, then unfold it. The unfolded pattern becomes a circle when it’s flattened out. What can we say about the angles of the crease pattern, merely from the knowledge that it came from a flatfoldable form? We can do this for any vertex of any folded form, and we call the resulting circular crease pattern the vertex crease pattern. The angular region between consecutive pairs of folds is a sector; the angle of each sector is, of course, the sector angle, already defined. Each sector in the crease pattern appears in the folded form and vice versa, and they are connected to each other in the same way in both the crease pattern and folded form; that is, sector θ 1 is connected to sector θ 2 , which is connected to sector θ 3 , and so forth. However, while in the crease pattern the sectors are all white side up and are counterclockwise ordered, in the folded form, some of the sectors are white side up while others are colored side up, and this property has important ramifications. Consider a circular folded vertex: its crease pattern and its folded form are illustrated in Figure 1.11, in which I have distorted the folded form so that all of the circular edges are visible, and I have assigned a consistent direction to each sector angle. That direction is indicated by a tiny black arrow in the crease pattern 18 ........CHAPTER 1. VERTICES 3 4 q3 6 2 q2 3 q1 q4 5 4 2 1 q6 q5 Figure 1.11. q3 q1 q5 6 1 5 Left: a vertex crease pattern with sector arcs assigned a direction. Right: the folded form of the vertex. and folded form, and it points consistently; that is, the arrow for sector 1 (angle θ 1 ) points from fold 1 toward fold 2, and so on, all the way around the circle. By following the arrows around the circular arcs, we traverse a complete circle in the crease pattern. By following the same path in the folded form, we no longer traverse a circle, but we still follow a closed path. Now, in the crease pattern, all of the directed sector angles run counterclockwise and must add up to a full circle, so we have an obvious relation on the sector angles, which can be generalized for N crease lines and sector angles: θ 1 + θ 2 + . . . + θ N = 360◦ . (1.3) If the sector angles around every vertex sum to 360◦ , then the vertex is said to be developable. For a planar crease pattern, this condition on the sector angles must hold for every vertex in the interior of the paper. In the folded form, though, some of the sector angles are turned over, and the directed arcs of those sectors run clockwise, rather than counterclockwise. Since they all connect up in a closed loop, the sum of the clockwise sector angles must be equal to the sum of the counterclockwise angles, so that as you traverse the loop, you end up in the same place that you started. As we go through the sectors in order, we see that they alternate: whiteup, colorup, whiteup, colorup, and so forth; and so all of the oddnumbered sector angles must have the same side up in the folded form, and all of the evennumbered sector angles must have the other side up in the folded form. In the example shown in Figure 1.11, the oddnumbered sectors are white side up and the evennumbered sectors are colored ........ CHAPTER 1. VERTICES 19 side up. So the total angle of the two sets must be equal; thus Õ Õ θi = θ i, (1.4) i odd i even and this relation must hold no matter where we start the numbering—or how many vertices are incident to the vertex. Thus, this brings us to a very powerful and general result, which applies to any vertex in the interior of the paper of a flatfoldable origami form, which is commonly stated as follows: Theorem 1 (KawasakiJustin Theorem). Let v be a vertex in an origami crease pattern, and let θ 1, θ 2, . . . , θ N be the angles between consecutive creases going around the vertex (N must be even). Then the vertex can fold flat if and only if θ 1 − θ 2 + θ 3 − θ 4 + . . . − θ N = 0. (1.5) The KawasakiJustin Theorem was described in the 1980s by Japanese mathematician Toshikazu Kawasaki [60], a prolific origami artist and mathematician, and Jacques Justin [54], a French mathematician who developed much of the mathematical theory of origami. (Actually, the theorem was proven even earlier, by S. A. Robertson in 1978 [104], but it is so widely associated with Kawasaki and Justin that I will continue to use the common name for it.) The number of creases incident on the vertex—the quantity N in the theorem—is called the degree of the vertex. Why must the degree be even? Well, as we travel around the vertex in the folded form, each time we cross a fold the paper switches from white side up to colored side up. In order to end on the same side where we started after going around the circle, we have to go through an even number of flips. Hence, the number of folds must be an even number. And why make the stipulation that the vertex must lie in the interior of the paper? If the vertex lies on the border, there is no way to create a closed loop, on which this result depends. The KawasakiJustin Theorem can be stated in many equivalent ways. One useful variation is the following: Theorem 2. Let v be a vertex in an origami crease pattern, and let θ 1, θ 2, . . . , θ N be the angles between consecutive creases going 20 ........CHAPTER 1. VERTICES around the vertex (N must be even). Then the vertex can fold flat if and only if θ 1 + θ 3 + θ 5 + . . . + θ N−1 = θ 2 + θ 4 + θ 6 + . . . + θ N = 180◦, (1.6) i.e., the sum of alternating angles around a flatfoldable vertex is equal to 180◦ . We call the condition on the angles of the KawasakiJustin Theorem the KawasakiJustin Condition. The KawasakiJustin Theorem and its variations can also be proved in many ways, and while the preceding demonstration was more of a handwaving exercise, we will encounter more rigorous formulations later on. We do note that the KawasakiJustin Condition is a necessary condition for flatfoldability, not sufficient; there are other conditions that must be satisfied as well, and we will encounter them shortly. The KawasakiJustin Theorem is one of the major tools in the arsenal of creating flatfoldable origami; many design rules boil down to ensuring that the KawasakiJustin Theorem is satisfied at every vertex of the crease pattern. We will have ample occasion to make use of the KawasakiJustin Theorem, so we will give it an abbreviation, KJT, which we will use later on. ? 1.2.2. Justin Ordering Conditions The mathematician Jacques Justin gave a concise set of mathematical conditions that must apply to the stacking order of a set of origami facets [56], which we will call the “Justin NonCrossing Conditions.” These conditions can be expressed formally and algebraically (as Justin did in [56], and as we will see later), but they are perhaps best appreciated pictorially, as shown in Figures 1.12–1.14. The Justin NonCrossing Conditions describe the three stacking order configurations that are valid and forbidden in a description of valid flatfolded origami. Imagine you are looking at a cross section of the paper. From top to bottom, they are the following: (a) If two creases overlap each other so that their facets overlap, then the facet pairs incident to the two creases cannot be interleaved, as in Figure 1.12. ........ CHAPTER 1. VERTICES 21 A B B A A B A B A A B B Figure 1.12. Left: allowed stacking orders for facets around two overlapping folded creases. Right: a forbidden stacking order. B B A A A B A B A A B B Figure 1.13. Left: allowed stacking orders for facets around a folded crease that overlaps an unfolded crease or facet. Right: a forbidden stacking order. (b) If a layer of paper overlaps a crease, it cannot lie between the facets incident to the crease, as in Figure 1.13. (c) If one facet lies above another on one side of an unfolded crease, it cannot lie below the other facet on the other side of the same line, as in Figure 1.14. A B Figure 1.14. A B A B Left: allowed stacking orders for facets around two overlapping unfolded creases. Right: a forbidden stacking order. 22 ........CHAPTER 1. VERTICES A B The Justin NonCrossing Conditions apply to the stacking order of the facets away from the crease, but the figures make it obvious why these configurations should be forbidden as a description of origami: they all involve the paper passing through itself. In all three cases, facet A passes through facet B at the dotted line. If we could see this sort of cross sectional picture of the folded form, it would be obvious whether there is a selfintersection and whether the picture describes a valid form of origami. The problem is that in the most common model of origami, we don’t have a picture in which the layers are spread apart like this. Usually we have just the crease pattern, and we have to find a stacking order that is consistent with the crease pattern—i.e., the folds drawn as mountain folds are ordered the way mountain folds are supposed to be, as shown in Figure 1.2, and the same for valleys—and that avoids any of the facet orderings shown in Figures 1.12–1.14. ? 1.2.3. Three Facet Theorem The KawasakiJustin Theorem is a theorem that stems from the nonstretchiness of the paper; more specifically, it follows from isometry, the fact that when we fold the paper, we do not change distances (or angular measures) on the folded form, as long as we’re measuring along the surface of the paper (and don’t jump between layers when they are stacked up). The Justin NonCrossing Conditions are different: they arise from the nonselfintersection requirement. They let us formulate a simple law that finds use surprisingly often in the design and analysis of folded structures. Let’s return to the simple crease pattern shown in Figure 1.5 and adjust the dimensions slightly, so that when folded, the two side flaps overlap not just each other, but also the opposite creases by just a bit, as shown in Figure 1.15. From the positions of the creases, we know what the silhouette of the folded form must be. The question to consider is, what are the possible crease assignments on the crease pattern, or equivalently, which layer lies on top of which in the folded form? As we have already noted, the stacking order among the overlapping facets provides a deeper description of the folded form than the crease pattern: given the former, we can work out the latter, while there may be more than one possible stacking order for a given crease assignment. ........ CHAPTER 1. VERTICES 23 C A B C A B Figure 1.15. Left: an unassigned crease pattern. Right: the silhouette of the folded form. So what are the possible stacking orders here? Could both A and C lie on top of B? If they did, then one of the two must be on top. If A is on top, then C lies between A and B, which means that C would slice through fold AB, violating case (b) of the Justin ordering conditions. Conversely, if C were on top, then A would lie between B and C, and A would slice through fold BC, also violating case (b). In either case, the paper intersects itself. A similar argument would apply if both A and C lay below B. Thus, we can set some constraints on layer order in this situation: Theorem 3 (Three Facet Theorem). Given three adjacent facets A, B, and C, where in the folded form facet A overlaps crease BC and flap C overlaps crease AB, facets A and C must lie on opposite sides of facet B. The proof of this theorem (henceforth, TFT) comes simply from considering all of the possible arrangements of the three facets: only the ones with A and B on opposite sides of B avoid selfintersection. By considering the relationship between facet order, twocoloring, and crease direction, we can establish a similar law that relates to crease assignment: Theorem 4 (Three Facet Crease Assignment). Given three adjacent facets A, B, and C, where in the folded form facet A overlaps crease BC and flap C overlaps crease AB, creases AB and BC must have opposite parity. This twocrease, threefacet arrangement is the simplest configuration where nonselfintersection plays a role in determining whether a crease pattern is flatfoldable or not. There are far more complex arrangements where selfintersection issues matter, and we will encounter many of them. 24 ........CHAPTER 1. VERTICES q3 q2 Figure 1.16. q1 Three consecutive sectors of a vertex, with θ 2 < θ 1 and θ 2 < θ 3 . ? 1.2.4. BigLittleBig Angle Theorem One of the simpler arrangements of three facets where TFT plays a role is the case where the two creases share a common vertex, as illustrated in Figure 1.16. In this case, the middle of the three angular sectors has a smaller angular measure than the two sectors to either side, so that, in the folded form, the conditions of TFT are satisfied. In general, for any two consecutive creases around a node, there are four possible crease assignments: • two mountain folds, • two valley folds, • mountain fold then valley fold, • valley fold then mountain fold. Following a terminology introduced by Palmer,2 we will call the angular sector where the two creases are of the same type an iso sector, whether they are both mountain or both valley, and we will refer to the two crease as iso creases. If the two creases differ, the sector is called an anto sector and the pair of creases are anto creases, as illustrated in Figure 1.17. q q q q Figure 1.17. iso 2 Private communication. anto Left: two iso sectors. Right: two anto sectors. ........ CHAPTER 1. VERTICES 25 q3 q2 q1 Figure 1.18. Three possible crease assignments for the two creases on either side of a smallest sector and the corresponding folded form. Top: a valid assignment, V M. Middle: a valid assignment, MV. Bottom: an invalid assignment, VV, which leads to a collision of the layers. q3 q2 q1 ! q3 q2 q1 Whatever else is happening with the other folds of the vertex, we can say one of two things definitively about the folds on either side of the middle sector: the creases must have opposite directions. As shown in Figure 1.18, they can be mountainvalley or valleymountain; but if both creases have the same assignment, then the wider sectors on either side of the short one collide as they try to fold past one another, as shown at the bottom of Figure 1.18. Thus, the only valid crease assignments for the two creases in this sector are the two anto assignments. And so, this gives another fundamental law of flatfoldability that was identified by Kawasaki [61, 60] and, a few years later, by Justin [56] (though I will use a name coined by Hull [51, p. 173]): Theorem 5 (BigLittleBig Angle (BLBA) Theorem). At any vertex, the creases on either side of any sector whose angle is smaller than those of its neighbors must have anto (opposite) crease assignment. We call the condition of the BLBA Theorem the BLBA Condition. It is important to note that this relation only holds for strict 26 ........CHAPTER 1. VERTICES inequality: the sector angle must be absolutely smaller than its neighbors to force the anto condition. Because this situation turns up fairly often, we will give it a corresponding name: a BLBA sector is a sector at a vertex whose angle is strictly smaller than those of the sectors to either side. ? 1.2.5. MaekawaJustin Theorem The KawasakiJustin Theorem deals with the sector angles, but not with the fold types, and follows purely from isometry of the paper. A second property addresses the fold types—mountain/valley status—themselves. As with the previous section, we will give a “plausibility argument” here, rather than a formal proof. Consider a folded vertex, like the example shown in Figure 1.19, oriented with the vertex at the bottom and white side on the outside. There must be a counterclockwisemost crease coming out of the vertex, such as the one labeled A in the figure, and a clockwisemost crease, labeled B in the figure. (If there are two or more creases at the extremal positions, you can pick one of them arbitrarily). Now consider what happens as we move from crease A to crease B along the figure. Since all of the facets we see are white, we must start on a whiteup facet; similarly, we must end on a whiteup facet. What about what happens in between? Imagine what happens along the circular edge. Every time we encounter a mountain fold, the path makes a 180◦ turn to the left; every time we encounter a valley fold, the path makes a 180◦ turn to the right. We can make two or more mountain or valley folds in succession, but the paper can’t penetrate itself, which means that the paper edge can’t form a complete loop; instead, every turn that is made at some point needs to be unwound by a turn in the opposite direction. And so, traveling across the front of this cone from crease A to crease B, there must be the same number of mountain folds as valley folds. A B Figure 1.19. A folded vertex. ........ CHAPTER 1. VERTICES 27 Exactly the same argument applies to the back side, of course. So, looking at all of the folds at the vertex, the number of mountain folds and valley folds must be the same—except for the two folds at the edges, which, for a whiteoutside vertex, are both mountain folds. So there are two more mountain than valley folds. If, however, we had started with a coloroutside vertex, we would have ended up with two more valley than mountain folds. But those are the only two possibilities. And so we have a general law about fold directions that applies to any flat vertex, which is called the MaekawaJustin Theorem, in honor of Jun Maekawa, who first identified the relation, and Justin, who proved it [55]. It states the following: Theorem 6 (MaekawaJustin Theorem). For any flatfoldable vertex, let M be the number of mountain folds at the vertex and V be the number of valley folds. Then M − V = ±2. (1.7) That is, for any vertex, the number of mountain folds and valley folds at that vertex must differ by exactly 2. We’ll use the abbreviation MJT for the MaekawaJustin Theorem. We’ll call Equation (1.7) the MaekawaJustin Condition. The argument presented above implicitly assumes nonselfintersection of the paper, because the assignment of mountain fold to both creases A and B is based on the assumption that the facet that reaches the rightmost crease B is still whiteup—which is a big assumption. For example, we can imagine something mysterious going on in the middle of the arc, as shown in Figure 1.20, in which a colorup layer somehow gets in front of the whiteup layer. In this case, crease B becomes a valley fold and the MaekawaJustin Theorem would not hold. But this type of rearrangement of the layers does not happen; and we can, indeed, rely upon the MaekawaJustin Theorem to A B ? Figure 1.20. A mystery folded vertex. 28 ........CHAPTER 1. VERTICES hold at every interior vertex—a vertex in the interior of the paper. Like the KawasakiJustin Theorem, it does not necessarily hold for vertices on the border of the paper (and usually does not). A related corollary gives a property we have already seen: any flatfoldable vertex must have an even number of creases emanating from it. From Equation (1.7), M = V ± 2, (1.8) the total number of creases must be (V ± 2) + V = 2(V ± 1), (1.9) which is clearly even. The argument presented above appeals heavily to intuition and so isn’t really a proof; but it turns out that MJT follows readily from a wellknown theorem in spherical geometry, Girard’s Theorem, which we will eventually meet. A flatfoldable vertex can be classified by the relationship between the numbers of creases of each type. The crease type that there is more of is the majority type; the other is the minority type. Flatfoldable vertices whose majority type is mountain are said to be mountainlike vertices; otherwise they are valleylike vertices. With duo paper—paper that is colored on one side and white on the other—a mountainlike vertex will be white in the folded form, and a valleylike vertex will be colored. This can be seen, for example, in Figures 1.25 and 1.27–1.29. ? 1.2.6. Vertex Type While the MaekawaJustin Theorem specifies the number of mountain and valley folds around a vertex, it does not say anything about their relative order around the vertex. We can concisely describe the fold order around a vertex by constructing a word composed of Ms and Vs giving the fold types one encounters as one goes around the vertex (in counterclockwise direction, by convention). We call this the vertex type. Figure 1.21 shows an example vertex of type VVV M MV. Of course, the vertex type is not unique for a given vertex: it depends on where we start counting. The vertex in Figure 1.21 is also VV M MVV, V M MVVV, M MVVVV, and so forth. And it does not fully specify the vertex: we would also need sector angles for a complete specification of the crease pattern, and we ........ CHAPTER 1. VERTICES 29 V V V Figure 1.21. M A vertex of type VVV M MV. M V would need the stacking order to fully specify the folded form. However, it is a useful shorthand for describing vertices, and so we will find occasion to use it as we go forward. ? 1.2.7. Vertex Validity Suppose we have a vertex and crease assignment (and sector angles). Is it valid? It may satisfy KJT and MJT, but it could still force a layer intersection somewhere along the way as we try to bring all of the folds together. We can determine this using an efficient procedure developed by Hull [49, 50] and described by Demaine and O’Rourke [22, p. 207]. Consider first a flat vertex v with sector angles (θ 1, θ 2, . . . , θ N ) and imagine that we begin to fold it up. If it has a smallest sector angle θi , we would start with that sector; it must be anto, and so there are two possible crease assignments, either MV or V M for the two creases on either side. If we form those two creases (but only those), the paper would now form a cone as shown in Figure 1.22, because we have effectively “taken a bite” out of the vertex circle by making these two folds. Ignoring the fact that the cone no longer lies flat, we could, in fact, treat this as a new vertex in which the trio of sector angles θi−1, θi, θi+1 has been replaced by a single sector angle whose value is θi0 = θi−1 − θi + θi+1 . We call this process sector reduction.3 Beginning with the flat, creaseassigned vertex, we ask: “is there a BLBA sector?” Meaning, if there are multiple equal smallest sectors, does any one of them have the anto crease as3 Demaine et al. refer to this process as “crimping” [22, p. 194], although generally in origami, a zigzag fold through one or more layers like this is called a “pleat” [68, pp. 30–31]. To avoid ambiguity, I will give the procedure its own distinct name. 30 ........CHAPTER 1. VERTICES qi+1qi+qi1 qi qi qi+1 qiÐ1 Figure 1.22. Left: a conical vertex in which angle θ i is the smallest sector angle. Middle: the cone resulting from one of the two possible crease assignments. Right: the vertex after reduction. signment? If so, reduce that sector, i.e., drop the two creases and replace the sector angle trio θi−1, θi, θi+1 with a single sector of angle θi0 = θi−1 − θi + θi+1 . Repeat the process, always looking for the smallest remaining sector angles, until you are left with two equal sectors. If at any point in the procedure there was no smallest angle that was anto, then the vertex was not flatfoldable; otherwise, it is (and, if we kept track of the layer orders at each reduction, the final configuration provides a valid layerordered solution). Call this procedure the Single Vertex FlatFoldable Test (SVFFT). If a creaseassigned vertex satisfies KJT, MJT, and SVFFT, then it is guaranteed to be valid, i.e., there exists at least one folded form that does not selfintersect. We should note, though, that the valid layerordered solution found by this procedure is not guaranteed to be unique. A vertex crease assignment does not fully specify the folded form, i.e., does not always determine uniquely the stacking order of the layers, as shown in Figure 1.23, which presents two different folded forms for the same vertex crease assignment. That ambiguity can play a role in determining flatfoldability for multiple sets of vertices— which, of course, most origami crease patterns consist of—as we will presently see. The vertex reduction process and associated crease assignment counting formulas become rather complex for arbitrary vertices of high degree. Most of the crease patterns that arise in mathematical origami, however, tend to have relatively low vertex degree, and thus only a few special cases apply. Now that we know some general properties about vertices, let’s look at some specific types of vertices. ........ CHAPTER 1. VERTICES 31 Figure 1.23. Left: a creaseassigned vertex. Middle: one folded form for this vertex. Right: another folded form for this vertex. ? 1.3. Degree2 Vertices The smallest flatfoldable vertex is the degree2 vertex, a vertex that has two creases emanating from it. A generic degree2 vertex is shown in Figure 1.24, although we can immediately see from the preceding laws that this cannot be flatfoldable as drawn. Since alternating angles must sum to zero from KJT, we must have θ 1 = θ 2 = 180◦ . And since M + V = ±2 from MJT, either both creases are mountain or both are valley. Thus, the only two possibilities for a degree2 vertex are that (a) the two creases must be collinear and (b) they must both have the same crease assignment, as shown in Figure 1.25. One might well say that there is no vertex there; that this is just a single crease line. But, for completeness, we should recognize that the degree2 vertex is still a possible vertex, and while the restriction to flatfoldability forces us to just these two configurations, if we allow 3D folding and/or curved or bent facets, even the humble degree2 vertex (or even a degree1 vertex!) can q1 Figure 1.24. A degree2 vertex with sector angles θ 1 and θ 2 . 32 ........CHAPTER 1. VERTICES q2 Figure 1.25. The two possible flatfoldable configurations for a degree2 vertex. give rise to shapes of considerable beauty, as can be seen in the work of British artist Paul Jackson in Figure 1.26. ? 1.4. Degree4 Vertices The next smallest flatfoldable vertex is the degree4 vertex, a vertex that has four creases coming from it. This type of vertex shows up often in origami crease patterns, particularly in mathematical Figure 1.26. Singlecrease (and singlevertex) threedimensional folds by Paul Jackson. Left: “He Said, She Said,” two squares of wetfolded 450 gsm watercolor paper. Photo courtesy of the Eretz Israel Museum. Right: “Untitled One Crease Form,” one square of wet folded 350 gsm watercolor paper. Originally published in [52]. Used by kind permission. ........ CHAPTER 1. VERTICES 33 folding. In fact, there are very many origami crease patterns that are composed exclusively of degree4 vertices. The degree4 vertex also has some special properties, as we will see, and so we will spend a bit of time focusing on this particular creature. Although there are four sector angles and four creases, the four angles are not independently choosable. If we label the four sector angles θ 1 , θ 2 , θ 3 , and θ 4 , then they must, of course, sum to 360◦ : θ 1 + θ 3 + θ 2 + θ 4 = 360◦ . (1.10) But also, from the KawasakiJustin Theorem, θ 1 − θ 2 + θ 3 − θ 4 = 0. (1.11) So, combining these two relationships, we find that θ 1 + θ 3 = θ 2 + θ 4 = 180◦, (1.12) that is, opposite angles must sum to 180◦ . So, there are only two degrees of freedom; one can choose any two adjacent sector angles, and then the other two are their respective supplements. Next, according to the MaekawaJustin Theorem, there are only two possibilities for crease assignment to such a vertex: three mountains and one valley fold, or three valleys and one mountain fold. That means that the assignment of the four creases will be either M M MV, VVV M, or a cyclic permutation of one of these. But not all threeofoneandoneoftheother crease assignments are possible because, as we have seen, the sector angles themselves play a role in what crease assignments are possible. ? 1.4.1. Unique Smallest Sector If there is a unique smallest sector angle, then according to the BLBA Theorem, that sector angle must be anto, i.e., bounded by two creases of opposite assignment. The other two creases, then, must have the same assignment, and so its sector is iso. Thus, there are only four possible crease assignments for such a vertex. The four possibilities for such a degree4 vertex—which we will call a uniquesmallestsector vertex—are shown in Figure 1.27. We note that if there is a unique largest sector angle, then the sector opposite to it must be the smallest and must also be unique, so we could have easily named such vertices uniquelargestsector vertices instead. We can say something about the 34 ........CHAPTER 1. VERTICES Figure 1.27. The four possible crease assignments for a uniquesmallestsector degree4 vertex and their folded forms. crease assignment at that largest sector, as well. Since the sector opposite has an anto assignment (either MV or V M), and the vertex must have three of one type and one of the other (either VVV M or M M MV), the two creases bounding the largest sector must have the same type, either M M or VV, and therefore, it must be iso. Thus, we have a corollary of the BLBA Theorem, albeit one exclusive to degree4 vertices: Theorem 7 (Unique Largest Angle Theorem). In a degree4 vertex, if there is a unique largest sector angle, its crease assignment is iso. The important ingredient here is that the smallest (largest) sector angle be unique; the other two sector angles may or may not be equal to each other. (The figure shows an example where they are unequal.) ? 1.4.2. Two Consecutive Smallest Sectors If the smallest sector angle isn’t unique, there are more than four possible crease assignments. Let us first consider what happens if two consecutive sector angles are equal to each other but are CHAPTER 1. VERTICES ........ 35 Figure 1.28. The six possible crease assignments for a degree4 vertex with two consecutive smallest sectors and their folded forms. both smaller than the other two sector angles. Since opposite sector angles add to 180◦ , then the remaining two sector angles must also be equal to each other, and if the first two sectors were the smallest, then they must both have sector angles less than 90◦ while the remaining pair has angles greater than 90◦ . Filtering all possible crease assignments by the BLBA Theorem to weed out the impossible assignments reveals that there are exactly six possible crease assignments, shown in Figure 1.28. 36 ........CHAPTER 1. VERTICES This particular type of degree4 vertex arises regularly in origami patterns, and so we will give it its own name: a flatfoldable degree4 vertex with two consecutive smallest sectors is a symmetric bird’sfoot vertex (or just “bird’sfoot” vertex, for short), named for its resemblance to the arrangement of toes on (most) perching birds. What if the two smallest sectors are equal but not consecutive— i.e., they are opposite? Then, since they sum to 180◦ , they must be equal to 90◦ . But if they are the smallest sectors, then the other two sectors must both have angles greater than 90◦ , which—since those other two angles must also sum to 180◦ —is not possible. What if we consider three consecutive equal smallest sector angles? Then the first and third angle of this trio must be opposite angles, and since they sum to 180◦ , they (along with the one between) must all be equal to 90◦ . In this case, the fourth sector angle must also be equal to 90◦ , which brings us to the third possible configuration for a degree4 vertex. ? 1.4.3. Four Equal Sectors If all four sector angles are 90◦ , then the vertex has fourfold symmetry. We can choose either M M MV or VVV M assignment, and we can pick any one of the four creases to be the “odd” crease in this assignment. Thus, there are eight possible crease assignments for this most symmetric vertex, which we call a right degree4 vertex. The eight possibilities are shown in Figure 1.29, along with their folded forms. And that completes the enumeration of the possible assignments for a degree4 vertex. Although this might seem like overkill of pedantry, it is useful to have an explicit list of the possibilities when one is assigning creases to a more complicated crease pattern consisting of degree4 vertices: it converts the crease assignment problem to a finite combinatorial problem. For each vertex, based on the angles, we can identify the set of four, six, or eight possibilities that apply to each individual vertex. Crease assignment then consists of assigning MV status to each of the creases so that the creases at each vertex form one of the acceptable possibilities. ? 1.4.4. Constructing Degree4 Vertices The situation regularly arises in origami design that three of the four creases at a flatfoldable vertex are known and the fourth is to be found. An example is shown in Figure 1.30. ........ CHAPTER 1. VERTICES 37 Figure 1.29. The eight possible crease assignments for a right degree4 vertex and their folded forms. q3 = ? Figure 1.30. A partial degree4 vertex. The remaining crease is to be found. 38 ........CHAPTER 1. VERTICES q4 = ? q2 = 60¡ q1 =100¡ 3 3 3 4 2 1 2 2 1 1. Extend line 1 across the vertex and duplicate line 2. 1 2. Rotate the wedge between the extension and line 2 so that the edge along line 2 is lined up with line 3. 3. Erase the right side of the wedge; the left side is the desired fourth crease. Figure 1.31. Geometric construction of the missing crease at a degree4 vertex when three are known. We can, of course, solve this problem mathematically by solving for the unknown angles. In a degree4 flatfoldable vertex, opposite angles sum to 180◦ , so we must have that θ 3 = 180◦ − θ 1 = 80◦, θ 4 = 180◦ − θ 2 = 120◦ . (1.13) (1.14) But what if we are drawing the crease pattern directly? Do we have to stop drawing and measure the angles, compute the new angles, then mark them off on the vertex? No, as it turns out. There are several ways to construct the required fold directly from the three existing creases. Perhaps the easiest is brute force: if you cut out the vertex and form the first three creases in the right place, then press it flat, the remaining crease will form automatically in exactly the right place. But we can also construct the remaining crease geometrically without having to measure any angles, as shown in Figure 1.31. Computer drawing programs typically offer tools to rotate selected items about a selected point, and sometimes even offer “snaptoobject” options that will give precise geometric alignments, making the construction of Figure 1.31 both straightforward and precise. An alternative construction devised by Ilan Garibi4 accomplishes the same result, but using reflection of single lines rather than rotation of a wedge. It is illustrated in Figure 1.32. 4 Private communication. ........ CHAPTER 1. VERTICES 39 3 3 2 2 Figure 1.32. An alternative geometric construction of the missing crease at a degree4 vertex when three are known. 1 1 1. Reflect line 2 across line 1 and again across line 3. 2. The desired crease is halfway between the two reflected lines. In the last step of Garibi’s construction, one must find the angle bisector between two lines. Some programs can create this directly, but if not, there is a simple workaround, shown in Figure 1.33. This still requires the ability to rotate a line by 90◦ about its midpoint, but this is a function that is very commonly built into vector drawing software. Using tricks such as this, it is often possible to construct quite complex and sophisticated flatfoldable crease patterns without ever computing a single angle. However, as we will eventually see, when we enter the realm of 3D folding, computation is nearly unavoidable. 3 3 2 2 1 1. Draw a line between the two radii with endpoints on the same circle. 3 1 2. Rotate the line by 90¡ about its midpoint. 2 1 3. Extend the line to the vertex to get the fourth crease line. Figure 1.33. Constructing a bisector between two given lines at the center of a circle can be accomplished with a rotation. 40 ........CHAPTER 1. VERTICES ? 1.4.5. HalfPlane Properties While we’re on the topic of the rotational positioning of creases around a vertex, there are several interesting properties of vertices that all relate to halfplanes, some of which we’ve already seen. For example, for degree2 vertices the following holds: Theorem 8 (Degree2 Vertex HalfPlanes Theorem). In the crease pattern of a flatfoldable degree2 vertex, the creases divide the paper into two halfplanes. This is, of course, just another way of saying that the two creases must be collinear. Things get a bit more interesting with degree4 vertices. Theorem 9 (Degree4 Vertex HalfPlanes Theorem). In the crease pattern of a flatfoldable degree4 vertex, every halfplane contains at least one crease of the majority type. We already know from the MaekawaJustin Theorem that there must be three folds of one type and one of the other; Theorem 9 tells us that the three cannot be excluded from any given halfplane. In fact, more broadly, every halfplane contains either exactly one or exactly two of the majority creases, except for the special case where all four angles equal 90◦ , in which case there is a halfplane that contains all three (with two of them on the border of the halfplane). The folded form also has halfplane relations: Theorem 10 (Vertex Folded Form HalfPlanes Theorem). In the folded form of a flatfoldable vertex, every crease lies within a common halfplane. This result turns out to be useful in numerical analysis: when solving for folded forms numerically, the mathematical conditions can sometimes give rise to spurious solutions that can be weeded out by applying this property. Note that this is not restricted to degree4 vertices; it is a property of every flatfolded interior vertex. All of these are relatively easily proven, and their proofs are left as an exercise for the interested reader. Theorem 9 also generalizes to higherorder vertices and, as well, to not necessarily flatfoldable vertices [1]: CHAPTER 1. VERTICES ........ 41 Theorem 11 (Partially Folded Vertex HalfPlanes Theorem). In the crease pattern of any vertex for which all creases can be at least partially folded simultaneously, there cannot be both • a halfplane that contains no mountain folds, • a halfplane that contains no valley folds. ?? 1.5. Multivertex FlatFoldability Thus far, I have given several explicit conditions for a single creaseassigned vertex to be flatfoldable: the KawasakiJustin Theorem, the MaekawaJustin Theorem, and Kawasaki’s BigLittleBig Angle Theorem. For a simple degree4 vertex, we can enumerate directly the possible assignments, given the sector angles; we can count all possible assignments, and we can use sector reduction to test if a given assignment is valid. The situation becomes a lot more complicated, however, when we start to consider networks of creases that consist of multiple vertices. Since every crease has two vertices and each vertex may place conditions on all of its incident creases, there is a possibility for different vertices to place contradictory conditions upon sets of creases. Even beyond that, it is possible to find crease assignments that are consistent at every vertex, but that result in collisions between layers of paper that, on the crease pattern, are far removed from one another. A crease assignment for which each vertex considered in isolation is valid is called locally flatfoldable. As the name suggest, it ensures that individual vertices can fold flat but provides no guarantee that the entire crease pattern folds flat without selfintersection. In fact, determining global flatfoldability can be a very challenging problem indeed—one that we will, by and large, bypass in our design and analysis. ?? 1.5.1. Isometry Conditions and Semifoldability In general, for a crease pattern to be foldable in the real world, it must satisfy both isometry and injectivity conditions. The extension of isometry conditions from individual vertices to entire networks of creases is generally straightforward; the extension of injectivity conditions (nonselfintersection) can get very complex indeed. It is natural, then, in origami design to take on the easy 42 ........CHAPTER 1. VERTICES P Figure 1.34. A Justin path on a crease pattern. part first: we solve for a set of creases that gives the right shape (which addresses isometry), then we look for a crease assignment that allows that shape to be folded without selfintersection (which addresses injectivity). In his paper “Towards a Mathematical Theory of Origami” that introduced many of the theorems of origami [56], Jacques Justin introduced the concept of semifoldability: a crease pattern is semifoldable if it satisfies isometry conditions. Thomas Hull has introduced the notion of ghost paper to describe this concept: allowing a model to satisfy flatfoldability but possibly allowing the paper to pass through itself like a ghost passing through a wall. If we allow ghost paper, then we are only addressing isometry, or to use Justin’s terminology, we only address semifoldability. Justin introduced several conditions related to semifoldability that apply to full crease patterns, not just individual vertices. They are all based on a common concept: the notion of a simple closed path on the crease pattern that doesn’t pass thro