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The Mathematics of Origami

Cover

Cambridge University Press

Cambridge webpage for book

ISBN 978-1-009-68735-5. Publication Dec. 2025.

Animations
Templates
Open Problems
Articles of Interest

Animated GIFs

A click on a thumbnail will run an animated GIF in a new browser window/tab. To repeat the animation, refresh the browser window.

All animations were made with Mathematica.

Chapter 4: Flat Folding is Hard

Fig. 4.6

Wire Splitter Gadget (500K)

Fig. 4.15

Triangle Twist Splitter Gadget(1MB)

Chapter 5: Rigid Origami and Degree-4 Vertices

Fig. 5.1

Open Box (1.8MB)

Fig. 5.3

Loop Table (Instagram link)

Fig. 5.10

Miura map fold (1MB)

Fig. 5.12

Miura Tessellation 85deg (1.5MB)

Fig. 5.16

Miura-unit 60deg (700KB)

Fig. 5.22

SquareTwist-unit 1Vertex (1.3MB)

Fig. 5.23

SquareTwist 4Vertices (1.5MB)

Fig. 5.25

SquareTwist Tessellation (2.5MB)

Fig. 5.26

Octahedron Top (1MB)

Chapter 6: Origami Design

Fig. 6.6

Rabbit Ear Molecule (1Mb)

Chapter 7: Fold & 1-Cut

Fig. 7.16

Cube Flattening (1.5Mb)

Chapter 9: Self-Folding Origami

Fig. 9.6, 9.8

Degree-6 vertex folding (Mode 1) (1.2Mb)

Templates

Click on the thumbnail to download a PDF of the template.

Chapter 2: Stamp Folding

Fig. 2.13(a)

3 x 2 Map

Fig. 2.14 4 x 2 Map

Chapter 3:Single Vertex Folds

Fig. 3.2a, 3.4a

Degree-8 vertex

Fig. 3.3c Not Flat Foldable

Fig. 3.6 Local Min

Fig. 3.7 Local Min Tie

Fig. 3.8 Kawasaki's Theorem

Fig. 3.11b Kawasaki's: Recutting

Chapter 4: Flat Folding is Hard

Fig. 4.3

Not Flattenable

Fig. 4.5a

Splitter Gadget

Fig. 4.14a

TriangleTwist Gadget

Chapter 5: Rigid Origamie and Degree-4 Vertices

Fig. 5.4

Loop Table

Fig. 5.10

Miura 3x2 Tessellation

Fig. 5.19a

Square Twist (a): M^4

Fig. 5.19b

Square Twist (b): M^2V^2

Fig. 5.22

Square Twist Tessellation

Chapter 6: Origami Design

Fig. 6.5a

Waterbomb base

Fig. 6.5b

Waterbomb crease pattern

Fig. 6.14a

Sawhorse molecule

Fig. 6.15b

Pentagon template

Fig. 6.16c

Uniaxial crease pattern

Chapter 7: Fold & 1-Cut

Fig. 7.1

Houdini Star

Fig. 7.6

Nonconvex Pentagon

Fig. 7.8

Turtle Straight Skeleton

From How To Fold It

Turtle with perpendiculars

Chapter 8: Curved Crease Origami

Fig. 8.4

Vessica Piscis

Fig. 8.7c

Nested Squares

Fig. 8.8a

Nested Circles

Chapter 9: Self-Folding Origami

Fig. 9.1b

Waterbomb Tube

Chapter 10: Origamizer

Fig. 10.13a

Degree-4 Vertex Tuck

Open Problems

Circular Pleat
Finite Perpendiculars
Map Folding
Orthogonally Aligned Creases
P =? NP
Single-Vertex
Stamp Foldings

Articles of Interest

"Origami Patterns Solve a Major Physics Riddle." 6Oct2025. Quanta link
"The Stamp Folding Problem From a Mountain-Valley Perspective." Thomas C. Hull, Adham Ibrahim, Jacob Paltrowitz, Natalya Ter-Saakov, Grace Wang. Discrete Mathematics & Theoretical Computer Science, vol. 27:3, Combinatorics 8Oct2025. arXiv link.