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Boca Raton London New York

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Taylor & Francis Group, an informa business

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Library of Congress Cataloging-in-Publication 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) | Origami--Mathematics.
Classification: LCC QA166.8 .L36 2018 | DDC 516/.132--dc23
LC record available at https://lccn.loc.gov/2017030497
Visit the Taylor & Francis Web site at
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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 ? . . . . . . . . . .

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1.2.1 Kawasaki-Justin Theorem ? . . . .
1.2.2 Justin Ordering Conditions ? . . . .
1.2.3 Three Facet Theorem ? . . . . . . .
1.2.4 Big-Little-Big Angle Theorem ? . .
1.2.5 Maekawa-Justin Theorem ? . . . .
1.2.6 Vertex Type ? . . . . . . . . . . . .
1.2.7 Vertex Validity ? . . . . . . . . . .
Degree-2 Vertices ? . . . . . . . . . . . . .
Degree-4 Vertices ? . . . . . . . . . . . . .
1.4.1 Unique Smallest Sector ? . . . . . .
1.4.2 Two Consecutive Smallest Sectors ?
1.4.3 Four Equal Sectors ? . . . . . . . .
1.4.4 Constructing Degree-4 Vertices ? .
1.4.5 Half-Plane Properties ? . . . . . . .
Multivertex Flat-Foldability ?? . . . . . . .
1.5.1 Isometry Conditions and
Semifoldability ?? . . . . . . . . .
1.5.2 Injectivity Conditions and Non-Twist
Relation ?? . . . . . . . . . . . . .
1.5.3 Local Flat-Foldability Graph ?? . .

xv
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xviii

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46
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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 Flat-Foldability ? ? ? . . . .
1.6.9 Analytic versus Numerical ? ? ?
Terms ? . . . . . . . . . . . . . . . . .

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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 Miura-ori ? . . . . . . . . .
2.3.4 Miura-ori Variations ? . . .
2.3.5 Barreto’s Mars ? . . . . . .
2.3.6 Generalized Mars ? . . . . .
Partial Periodicity ?, ??, ? ? ? . . .
2.4.1 Yoshimura-Miura Hybrids ?
2.4.2 Semigeneralized Miura-ori ?
2.4.3 Predistortion ?? . . . . . .
2.4.4 Tachi-Miura Mechanisms ? .
2.4.5 Triangulated Cylinders ? . .

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2.4.6 Triangulated Cylinder Geometry ? ? ?
2.4.7 Waterbomb Tessellation ? . . . . . .
2.4.8 Troublewit and Pleats ? . . . . . . . .
2.4.9 Corrugations and More ? . . . . . . .
Terms ? . . . . . . . . . . . . . . . . . . . .

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3 Simple Twists
3.1
3.2

viii

........CONTENTS

Twist-Based 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
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67
70
73
79
80
80
85
88
90
93
101
106
114
117
121
126
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138
144
152
160
164
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184
190

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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 Closed-Back Twists ? . . . .
3.4.3 Rotation Angle of the Central Polygon ?
3.4.4 Iso-Area Twists ?? . . . . . . . . . . .
Twist Flat-Foldability ? . . . . . . . . . . . . .
3.5.1 Local Flat-Foldability ? . . . . . . . .
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 Higher-Order Irregular Twists ?? . . .
3.6.3 Cyclic Overlaps in Irregular Twists ?? .
3.6.4 Closed-Back Irregular Twists ?? . . . .
3.6.5 Open-Back 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 ?, ?? . . . . .

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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
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306
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311
312
312
316
CONTENTS

........

ix

4.6

4.7

Higher-Order 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 Higher-Order Polygon
Tiles ?? . . . . . . . . . . . . . . . .
4.6.5 Pathological Twist Tiles ? . . . . . .
4.6.6 Split-Twist Quadrilateral Tiles ? . . .
Terms ? . . . . . . . . . . . . . . . . . . . .

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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 ? ? ? . . .
Edge-Oriented Tilings ? . . . . . . . . . . .
5.3.1 Centered Twist Tiles ? . . . . . . . .
5.3.2 Offset Twist Tiles ? . . . . . . . . . .
k-Uniform Tilings ? . . . . . . . . . . . . . .
5.4.1 2-Uniform Tilings ? . . . . . . . . .
5.4.2 Two-Colorable 2-Uniform Tilings ? .
5.4.3 Higher-Order Uniform Tilings ? . . .
5.4.4 Periodic Tilings with Other Shapes ?
5.4.5 Grid Tessellations ? . . . . . . . . . .
Non-Periodic Tilings ?, ? ? ? . . . . . . . . .
5.5.1 Goldberg Tiling ? . . . . . . . . . . .
5.5.2 Self-Similar Tilings ? ? ? . . . . . . .
Terms ? . . . . . . . . . . . . . . . . . . . .

6 Primal-Dual Tessellations
6.1
6.2
6.3

x

........CONTENTS

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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 ?? . . . . . . . . . . . . . . .

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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 ? . . . . . . .
Half-Open Vertices ?? . . . . . . . . . . . .
Spherical Geometry ?? . . . . . . . . . . . .
A Degree-4 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 ?? . . . . . .
Non-Twist Folds ?? . . . . . . . . . . . . . .
7.7.1 General Meshes ?? . . . . . . . . . .
7.7.2 Quadrilateral Meshes ?? . . . . . . .
Non-Quadrilateral Meshes ? . . . . . . . . .
7.8.1 Forced Rigid Foldability ? . . . . . .
7.8.2 Non-Flat-Foldable Vertices ? . . . . .
Terms ? . . . . . . . . . . . . . . . . . . . .

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423
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475
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489
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495
498
500
506
508
512
515
515
518
528
528
530
533

CONTENTS

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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 Degree-4 Vertex ??
Sector and Fold Angles ?? . .
8.3.1 Osculating Plane ?? .
8.3.2 Binding Condition ??
8.3.3 Ruling Plane ?? . . .

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8.3.5 Ruling Angle ?? . . . . . . . . . . . .
8.3.6 Osculating Angle ?? . . . . . . . . . .
8.3.7 Adjacent Fold Angles ?? . . . . . . . .
8.3.8 Flat-Foldable and Straight-Major/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 Miura-ori 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 Degree-4 Vertex from Sector Elevation
Angles ? ? ? . . . . . . . . . . . . .
Discrete Space Curve ? ? ? . . . . . . . . . .

535
536
536
538
541
543
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547
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561
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615
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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 ? . . . . . . . . . . . . . . . . .

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10 Rotational Solids
10.1
10.2
10.3

10.4
10.5
10.6
10.7

Three-Dimensional Twists ?, ?? . . .
10.1.1 Puffy Twists ? . . . . . . . . .
10.1.2 Folding a Sphere ?? . . . . .
Thin-Flange Algorithm ? ? ? . . . . .
Thick-Flange Structures ?, ? ? ? . . .
10.3.1 Mosely’s “Bud” ? . . . . . . .
10.3.2 Thick-Flange 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
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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
paper-folding, is. Everyone knows, because they have seen the
well-known Japanese crane, or tsuru, an international symbol
of peace. Or they have folded schoolyard paper-folding—boats,
bangers, and cootie-catchers, 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 well-known 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 non-representational decorative paper-folding, both within
xv

the world of Japanese origami and coming from many disparate
fields of endeavor outside of the Japanese tradition, ranging from
napkin-folding of the 15th century in Europe to early-20th-century
Bauhaus architecture to late-20th-century computational geometry and mathematics.
It is this latter field of non-representational 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 three-dimensional. What these new folded-paper forms had
in common were that they were non-representational and often
highly geometric.
The origami tessellation explorers also discovered something
else: mathematical and non-representational 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 paper-folding. 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, single-sheet 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 fast-growing 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 (equation-solving), 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 step-by-step 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
paper-folding. 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 non-stretchy sheet-like
material, manipulated in three dimensions, with few or no cuts.
Mathematical folding doesn’t require that you use paper—in fact,
in real-world 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 hand-eye
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 three-dimensional (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:
• Non-stretchy paper. The folded shape is a 3D deformation of a planar surface that does not appreciably
stretch (or compress) in any direction.
• Non-self-intersection. The paper cannot intersect itself
in the folded form, or in any intermediate stage.
Any mathematical description of paper-folding must include
these two properties in some way or another. These two
properties—non-stretchiness and non-self-intersection—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 non-stretchiness and non-self-intersection. 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 non-self-intersection. 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
zero-thickness 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;
squash-fold; petal-fold; and so forth, where each term (“squashfold,” “petal-fold,” etc.) refers to a specific manipulation involving
a small number of folds at a time. While this linear step-by-step
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
Flat-Foldable 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 paper-folding.
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 flat-foldable origami. Such a
model is, of course, an approximation of reality; there is no such
thing as zero-thickness 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 one-to-one 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 step-by-step 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 paper-folding 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 mid-20th 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 (dot-dot-dash). These conventions are now widely established in step-by-step 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 step-by-step
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 step-by-step 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
color-blind 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 color-blind 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 mountain-valley 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
step-by-step 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 self-intersection.
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 long-range
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 non-local interactions between
its constituent parts. The foldability of the pattern, and/or the
validity of its crease assignment, can depend on relationships
between far-flung 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:
flat-foldable 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 flat-folded 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 flat-folded 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 crease-assigned (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 Maekawa-Justin 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 white-up
(white side facing the viewer) and the other facet must be color-up
(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 white-up facet to a color-up facet or vice versa.
Thus, for any flat origami crease pattern, if we color the facets
according to whether they are white-up or color-up, this coloring
has the property that no two facets of the same color meet along
a common fold line. This is called a two-coloring of the crease
pattern. Every flat origami crease pattern can be two-colored
(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 white-up or color-up provides
such a coloring.
For any given two-colorable pattern, there are only two possible two-colorings: 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 two-colorings of that pattern automatically gives a map of which facets are white-up and which
are color-up 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 flat-foldability 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. Kawasaki-Justin Theorem
The first property of vertices in a crease pattern ultimately derives
from the condition of non-stretchiness (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 circle-cut 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
flat-foldable 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: white-up, color-up,
white-up, color-up, and so forth; and so all of the odd-numbered
sector angles must have the same side up in the folded form, and
all of the even-numbered sector angles must have the other side
up in the folded form.
In the example shown in Figure 1.11, the odd-numbered sectors are white side up and the even-numbered 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 flat-foldable
origami form, which is commonly stated as follows:
Theorem 1 (Kawasaki-Justin 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 Kawasaki-Justin 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 Kawasaki-Justin 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 flat-foldable vertex is
equal to 180◦ .
We call the condition on the angles of the Kawasaki-Justin
Theorem the Kawasaki-Justin Condition.
The Kawasaki-Justin Theorem and its variations can also be
proved in many ways, and while the preceding demonstration was
more of a hand-waving exercise, we will encounter more rigorous
formulations later on.
We do note that the Kawasaki-Justin Condition is a necessary
condition for flat-foldability, not sufficient; there are other conditions that must be satisfied as well, and we will encounter them
shortly.
The Kawasaki-Justin Theorem is one of the major tools in the
arsenal of creating flat-foldable origami; many design rules boil
down to ensuring that the Kawasaki-Justin Theorem is satisfied at
every vertex of the crease pattern. We will have ample occasion
to make use of the Kawasaki-Justin 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 Non-Crossing
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 Non-Crossing Conditions describe the three stacking order configurations that are valid and forbidden in a description of valid flat-folded 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 Non-Crossing 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 self-intersection 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 Kawasaki-Justin Theorem is a theorem that stems from the
non-stretchiness 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 Non-Crossing
Conditions are different: they arise from the non-self-intersection
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
self-intersection. By considering the relationship between facet
order, two-coloring, 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 two-crease, three-facet arrangement is the simplest configuration where non-self-intersection plays a role in determining
whether a crease pattern is flat-foldable or not. There are far more
complex arrangements where self-intersection 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. Big-Little-Big 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 mountain-valley or
valley-mountain; 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 flat-foldability 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 (Big-Little-Big 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. Maekawa-Justin Theorem
The Kawasaki-Justin 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 counterclockwise-most crease
coming out of the vertex, such as the one labeled A in the figure,
and a clockwise-most 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 white-up facet; similarly, we must
end on a white-up 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 white-outside vertex, are both mountain
folds. So there are two more mountain than valley folds.
If, however, we had started with a color-outside 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 Maekawa-Justin Theorem, in honor of Jun
Maekawa, who first identified the relation, and Justin, who proved
it [55]. It states the following:
Theorem 6 (Maekawa-Justin Theorem). For any flat-foldable 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 Maekawa-Justin Theorem. We’ll call
Equation (1.7) the Maekawa-Justin Condition.
The argument presented above implicitly assumes non-selfintersection 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 right-most crease B is still white-up—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 color-up layer somehow gets in front of the
white-up layer. In this case, crease B becomes a valley fold and
the Maekawa-Justin Theorem would not hold.
But this type of rearrangement of the layers does not happen;
and we can, indeed, rely upon the Maekawa-Justin 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 Kawasaki-Justin 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 flat-foldable 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 well-known theorem in spherical geometry, Girard’s Theorem,
which we will eventually meet.
A flat-foldable 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. Flat-foldable vertices whose majority type is mountain are
said to be mountain-like vertices; otherwise they are valley-like
vertices. With duo paper—paper that is colored on one side and
white on the other—a mountain-like vertex will be white in the
folded form, and a valley-like 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 Maekawa-Justin 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, crease-assigned 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+1-qi+qi-1

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 flat-foldable;
otherwise, it is (and, if we kept track of the layer orders at each
reduction, the final configuration provides a valid layer-ordered
solution). Call this procedure the Single Vertex Flat-Foldable
Test (SVFFT). If a crease-assigned 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 self-intersect.
We should note, though, that the valid layer-ordered 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 flat-foldability 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 crease-assigned vertex.
Middle: one folded form for this vertex.
Right: another folded form for this vertex.

?

1.3. Degree-2 Vertices
The smallest flat-foldable vertex is the degree-2 vertex, a vertex
that has two creases emanating from it. A generic degree-2 vertex
is shown in Figure 1.24, although we can immediately see from
the preceding laws that this cannot be flat-foldable 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 degree-2 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 degree-2 vertex is still a possible vertex, and
while the restriction to flat-foldability forces us to just these two
configurations, if we allow 3D folding and/or curved or bent facets,
even the humble degree-2 vertex (or even a degree-1 vertex!) can

q1

Figure 1.24.
A degree-2 vertex with
sector angles θ 1 and θ 2 .

32

........CHAPTER 1. VERTICES

q2

Figure 1.25.
The two possible
flat-foldable
configurations for a
degree-2 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. Degree-4 Vertices
The next smallest flat-foldable vertex is the degree-4 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.
Single-crease (and single-vertex) three-dimensional folds by Paul Jackson.
Left: “He Said, She Said,” two squares of wet-folded 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 degree-4 vertices. The degree-4 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 Kawasaki-Justin 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 Maekawa-Justin 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 three-of-one-and-one-of-the-other 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 degree-4 vertex—which we will
call a unique-smallest-sector 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 uniquelargest-sector vertices instead. We can say something about the

34

........CHAPTER 1. VERTICES

Figure 1.27.
The four possible crease assignments for a unique-smallest-sector degree-4 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 degree-4 vertices:
Theorem 7 (Unique Largest Angle Theorem). In a degree-4 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 degree-4 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 degree-4 vertex arises regularly in
origami patterns, and so we will give it its own name: a flatfoldable degree-4 vertex with two consecutive smallest sectors
is a symmetric bird’s-foot vertex (or just “bird’s-foot” 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 degree-4 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
degree-4 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 degree-4 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 degree-4 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 Degree-4 Vertices
The situation regularly arises in origami design that three of the
four creases at a flat-foldable 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 degree-4 vertex and their folded forms.

q3 = ?

Figure 1.30.
A partial degree-4
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 degree-4 vertex when three are known.

We can, of course, solve this problem mathematically by solving for the unknown angles. In a degree-4 flat-foldable 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
“snap-to-object” 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 degree-4 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. Half-Plane 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 half-planes, some of which we’ve already seen.
For example, for degree-2 vertices the following holds:
Theorem 8 (Degree-2 Vertex Half-Planes Theorem). In the crease
pattern of a flat-foldable degree-2 vertex, the creases divide the
paper into two half-planes.
This is, of course, just another way of saying that the two
creases must be collinear. Things get a bit more interesting with
degree-4 vertices.
Theorem 9 (Degree-4 Vertex Half-Planes Theorem). In the crease
pattern of a flat-foldable degree-4 vertex, every half-plane contains at least one crease of the majority type.
We already know from the Maekawa-Justin 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 half-plane 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 half-plane that contains all three (with two of them on the border
of the half-plane).
The folded form also has half-plane relations:
Theorem 10 (Vertex Folded Form Half-Planes Theorem). In the folded form of a flat-foldable vertex, every crease lies within a common half-plane.
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 degree-4 vertices; it is a property of every flat-folded 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 higher-order vertices and, as well, to not necessarily
flat-foldable vertices [1]:
CHAPTER 1. VERTICES

........

41

Theorem 11 (Partially Folded Vertex Half-Planes Theorem). In the
crease pattern of any vertex for which all creases can be at least
partially folded simultaneously, there cannot be both
• a half-plane that contains no mountain folds,
• a half-plane that contains no valley folds.
??

1.5. Multivertex Flat-Foldability
Thus far, I have given several explicit conditions for a single
crease-assigned vertex to be flat-foldable: the Kawasaki-Justin
Theorem, the Maekawa-Justin Theorem, and Kawasaki’s BigLittle-Big Angle Theorem. For a simple degree-4 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 flat-foldable. 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 flat-foldability 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 (non-self-intersection) 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 self-intersection (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 flat-foldability 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