How To Fold It:

The Mathematics of Linkages, Origami, and Polyhedra

Cambridge University Press

(Published 2011)

CoverSmall

Cambridge University Press link: Cambridge. Amazon link: Amazon.

Cambridge Description: What do proteins and pop-up cards have in common? How is opening a grocery bag different from opening a gift box? How can you cut out the letters for a whole word all at once with one straight scissors cut? How many ways are there to flatten a cube? You can answer these questions and more through the mathematics of folding and unfolding. From this book, you will discover new and old mathematical theorems by folding paper and find out how to reason toward proofs. With the help of 200 color figures, author Joseph O'Rourke explains these fascinating folding problems starting from high school algebra and geometry and introducing more advanced concepts in tangible contexts as they arise. He shows how variations on these basic problems lead directly to the frontiers of current mathematical research and offers ten accessible unsolved problems for the enterprising reader. Before tackling these, you can test your skills on fifty exercises with complete solutions.

Nearly 200 color figures and 50 exercises, with complete solutions
Written in accessible language and discusses tangible topics which render abstract mathematics concrete
Presumes only high-school algebra and geometry

Now translated into Japanese by Ryuhei Uehara: http://www.jaist.ac.jp/~uehara/books/howtofoldit/ : Japanese cover

Errata (Updated 9 May 2012)

Last Update to this page:

Closed Open Problems

The book includes descriptions of ten open problems (see the Index under open problem, p. 176). Inevitably, some will be resolved as time passes and before a Second Edition is prepared. Here I will maintain a list of those Open Problems in the book which have been closed, or on which some notable advance has been achieved.

Open Problem: Flattening Polyhedra (p. 83)

The special case of convex polyhedra has been solved (Yes, they can all be continuously flattened!) Abel, Z., E. D. Demaine, M. L. Demaine, J.-i. Itoh, A. Lubiw, C. Nara, and J. O’Rourke (2014). Continuously flattening polyhedra using straight skeletons. In Proc. 30th Annual Symposium on Computational Geometry (SoCG), pp. 396–405. Association for Computing Machinery. Link to download PDF.
Open Problem: Planar Signing (p. 37): Is there a planar linkage that signs your name?

Yes! Zachary Abel. "On folding and unfolding with linkages and origami." Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mathematics, 2016. MIT link.
From Abstract: "we prove a strong form of Kempe's Universality Theorem for linkages that avoid crossings."

See also the longer list of Closed/Open Problems associated with the more advanced monograph, Geometric Folding Algorithms: Linkages, Origami, Polyhedra.

MathFest 2012 MiniCourse (2-4 August 2012) PowerPoint slides (27MB download)

Below appears some supplementary material, organized by chapter. Click on links to jump to that chapter's material:

Chapter 3 link
Chapter 4 link
Chapter 5 link
Chapter 6 link
Chapter 7 link
Chapter 8 link
Chapter 9 link

Chapter 3: Protein Folding and Pop-up Cards

Animation by Akira Nishihara. (Click on image to see animated GIF): Nishihara web page.


Fig. 3.14 (p.51)	Uncurling of a unit 90°-chain as it stretches to its maxspan configuration. [Video by Julia Patterson]

Final frame of animation

The video is a first attempt at animating the uncurling of the spiral unit 90°-chain. The jittering near the end of the stretch is under investigation...

Chapter 4: Flat Vertex folds

Templates

Click on thumbnail images to bring up 8½ x 11" PDF ready for printing.

Where	Description	Templates (Color)	Templates (B&W)
Fig. 4.4	Degree-6 vertex
Fig. 4.7	Degree-6 non-flat
Fig. 4.8	Maekawa-Justin Theorem
Fig. 4.14	Map puzzle

Chapter 5: Fold and One-Cut

Templates

Click on thumbnail images to bring up 8½ x 11" PDF ready for printing.

Where	Description	Templates (color)
Fig. 5.2	Square
Fig. 5.3	Star
Fig. 5.5	Rectangle
Fig. 5.7	Irregular Triangle
Fig. 5.9	Square & Rectangle
Fig. 5.11	Letter 'A' [This corrects one incorrectly labeled crease in Fig.5.11(c).]
	Letter 'A' with hole
Fig. 5.12	Turtle	Step-by-step instructions

Chapter 6: The Shopping Bag Theorem

Hanegraff hinged dissection, corresponding to Fig. 6.4. Video created by Thomas Kent.

Chapter 7: Dürer's Problem: Edge Unfolding

Here is a dynamic version of Figure 7.11: The Latin cross unfolding of the cube.

Cube to Latin cross. [Video by Katie Park]

Here is an edge-unfolding of a cube to a 'Z'-shape, and then a refolding of the 'Z' to double-covered parallelogram. (This example is not in the book, but related.) All the Platonic solids but the dodecahedron have similar edge-unfoldings and refoldings to a parallelogram. Template below in Chapter 9 section.

Unfolding cube to 'Z'.	Folding 'Z' to parallelogram.

[Videos by Katie Park]

Templates

Where	Description	Templates
Ex. 7.7, Fig. 7.22(a)	Star Unfolding
Fig. 7.21(b)	2 × 1 × 1 Box

Chapter 8: Unfolding Orthogonal Polyhedra

These four videos show animations of the orthogonal terrain algorithm described in Section 8.1, especially Figures 8.2 and 8.4. For more detail, see my note, "Unfolding Orthogonal Terrains."

10 x 10 Terrain, Example 1	10 x 10 Terrain, Example 2


10 x 10 Top only (dynamically rescaled)	20 x 20 Terrain

The video below illustrates the basic construction that underlies the algorithm described in Section 8.4 Above & Beyond, especially Figure 8.8. For more detail, see the paper, "Epsilon Unfolding of Orthogonal Polyhedra."

Helical unwrapping one box to a staircase shape. [Video by Robin Flatland and Ray Navarette]

Chapter 9: Folding Polygons to Convex Polyhedra

Templates

Where	Description	Templates
(Not in book)	Which folds to cube?
(Not in book)	Cube Z → Octahedron
Fig. 9.6	Square perimeter-halving.
(Not in book. Related to Fig. 9.2. See video above.)	Cube Z to Parallelogram
Fig. 9.9	Latin cross to Tetrahedron
Fig. 9.10	'SoCG' polygon: Folds to what?