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

Joseph O'Rourke

Cambridge University Press 2011
A book review by Danny Yee © 2019
How to Fold It is an exploration of some of the mathematics associated with folding, looking at robot arms and similar linkages, at origami, and at polyhedra. It is accessible with only basic secondary school mathematics, but the material is not covered in typical undergraduate courses; much of it was new to me. It's also a lot of fun!

"Linkages" looks at how to work out the area reachable by a robot arm in two dimensions, and considers extensions to three dimensions and ruler-folding. It then considers a problem that originated in engineering, using straight-line linkages to constrain a piston to move in a straight line: Watt came up with an approximation, but it wasn't till 1864 that Peaucellier found a linkage with an exact straight-line motion. Also touched on are some common features of protein folding and pop-up cards, modelled as fixed angle chains.

"Origami" begins with flat-vertex folds, concentrating on what happens at a single vertex, but leading up to unsolved problems in map-folding. It then introduces the Fold and One-Cut Theorem — that any shape made up of straight lines can be cut out, after folding, with a single cut — and the Shopping Bag theorem — that collapsing a typical paper grocery bag requires bending its flat faces.

"Polyhedra" begins with a problem that goes back to Dürer but is still open, asking whether any convex polyhedron can be cut along edges and unfolded onto a plane without overlapping. It presents some results that have been achieved when this is restricted to orthogonal polyhedra. And then it looks at going the other way, at folding polygons to polyhedra, and introduces a theorem of Alexandrov.

This is all pitched so as to be at least potentially accessible to secondary school students, though their teachers may be a more obvious audience. Indeed not much more than an elementary understanding of angles, algebra and arithmetic is assumed: there's no trigonometry, just one passing reference to Euclidean geometry, simple vectors are explained when introduced, and so forth. The challenge it poses is in spatial visualisation rather than anything else.

How to Fold It does attempt to build up, however, broader mathematical knowledge: it includes proofs, introduces terminology and notation conventions, and tries to teach mathematical ways of thinking. There are also hints at more advanced ideas — passing mentions of PSPACE and NP-complete complexity, for example, and statements of unsolved problems — and there are five pages of annotated "further reading" suggestions.

There are some mentions of applications and history, with illustrations, but those are not a primary focus. Similarly, there are some suggestions for practical constructions, but those are not central to the exposition. There's a small but useful selection of exercises: basic "Practice" reiteration of material, some "Understanding" testing conceptual grasp, and a few significantly harder "Challenge" problems. Full answers to these are provided.

A motivated student could use How to Fold It by themselves, but it would also be an ideal basis for an extra-curricular workshop or course. Every secondary school library should have a copy, in any event.

July 2019

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%T How To Fold It
%S The Mathematics of Linkages, Origami, and Polyhedra
%A O'Rourke, Joseph
%I Cambridge University Press
%D 2011
%O paperback, glossary, exercises, answers, index
%G ISBN-13 9780521145473
%P 177pp