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

Joseph O'Rourke
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Alex Bogomolny
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How To Fold It is a splendid little book that grew out of the more demanding monograph Geometric Folding Algorithms by E. Demaine and J. O’Rourke. According to the author, both of Demaine and O’Rourke had experience presenting aspects of the material in the monograph at various educational levels — from fifth grade to high school. Both found that the tangibility of the topics made them accessible through physical intuition. The book is — in my view — a very successful attempt to capitalize on the readers’ intuition in introducing several beautiful facets of contemporary mathematics.

The book, however, should not be perceived merely as a collection of hands-on activities. The author successfully weaves a synergetic combination of do-it-yourself activities and see-why-it-works theory. The do-it-yourself portions could be performed by middle schoolers; and so might some of the mathematical expositions which never require experience beyond common high school curriculum. The author goes to a considerable length to make the book accessible to a broad audience. Some of the ancillary mathematics — vectors, mathematical induction, triangle inequality, the theorem of the incidence of the angle bisectors in a triangle, convexity — are conveniently framed in blue boxes and stand out to be easily found or skipped, depending on the reader’s level.

On the whole, the book is very well illustrated with high quality diagrams and photographs. My only peeve is with the manner the author adopted for denoting valley and mountain folds in the 2D — origami — part of the book. Usually, these are distinguished by the lines drawn with different patterns (dashed, dotted, or a combination). The book makes a distinction by using two different colors — red and green. This is unfortunate. The sad fact is that about 8% of male population are red-green color blind. I, for one, could hardly tell one kind of the folds from the other. From experience, I would be far more comfortable with the lines of different thickness.

The book consists of three parts that deal with folding of objects loosely defined as 1-, 2-, and 3-dimensional. The subject of the first is linkages, that of the second origami, while the third part deals with polyhedral nets and related questions. To give a little more extensive description, the first part contains discussions on the reachability region of robot arms, linkages transforming between curved and straight line motions, pantograph and protein folding. The second part takes up flat foldings and the way to produce various shapes with a single straight line cut (after folding of course). Here the reader meets results that are not new but are still not widely known: theorems of Maekawa/Kawasaki/Justin and the hugely entertaining “Shopping Bag” theorem.

The third part starts with a problem posed yet by Albrecht Dürer: does every convex polyhedron have a net? At first glance, the answer to the 500 year old problem seems obvious; O’Rourke explains why it is not — in general. Some families of polyhedra do have nets, though. The author proves that the so-called orthogonal polyhedra — likely the invention of some urban planner — do have nets. The final topic is the folding of polygons into polyhedra. Say, what can you do with the Latin Cross? (The author can fold 23 different shapes: the cube and several of tetrahedra, pentahedra, hexahedra, and octahedra.)

There are three chapters in each of the three parts, each ending with an “Above & Beyond” section that gives a glimpse of deeper or less specialized results. There are also 50 exercises, with complete solutions at the end of the book. The author also highlights a great variety of open problems. As the author observes, the topic of foldings is attractive in that many of its unsolved problems are accessible to the novice and might be solved by just the right clever idea. He’d appreciate being informed if you crack one of them. This you can do through the book’s web site The site contains videos, some extra material, and a short errata.

Alex Bogomolny is a freelance mathematician and educational web developer. He regularly works on his website Interactive Mathematics Miscellany and Puzzles and blogs at Cut The Knot Math. Follow Alex on twitter

Part I. Linkages: 1. Robot arms
2. Straight-line linkages and the pantograph
3. Protein folding and pop-up cards
Part II. Origami: 4. Flat vertex folds
5. Fold and one-cut
6. The shopping bag theorem
Part III. Polyhedra: 7. Durer's problem: edge unfolding
8. Unfolding orthogonal polyhedra
9. Folding polygons to convex polyhedra
10. Further reading
11. Glossary
12. Answers to exercises
13. Permissions and acknowledgments.