Polyhedra are among the most beautiful objects in mathematics. Everyone recognizes their physical beauty, and an understanding of the mathematics behind them helps us appreciate them even more.
This remarkable book goes far beyond the superficial, providing a solid and fascinating account of the history and mathematics of polyhedra, especially regular polyhedra. It is likely to become the classic book on the topic.
Polyhedra are much more complicated than polygons. For example, as we learn in Chapter 1, any polygon can be dissected into triangles that can be reassembled to form a square. Hence, the area of a polygon is an elementary concept. Polyhedra, on the other hand, cannot always be dissected and reassembled to form another shape. This is the crux of Hilbert's Third Problem.
We all know the five Platonic solids, and some people regard them as the only "regular" polyhedra. But what properties make them "regular"? Several different definitions have been used, and they give rise to several different families of polyhedra, each with a claim to being regular. For example, if you ask for face-, edge- and vertex-transitivity, you might not even get a finite polyhedron, as we learn in Chapter 2.
In Chapter 3, Cromwell leads us through a history of polyhedra, from Plato and Archimedes, through the Renaissance and the ideas of perspective. In Chapter 4 we learn of Kepler's work with star-like polyhedra, and of Descartes' analytical work on the subject. We also see some wild extensions of the idea of a polyhedron, including self intersections, and polyhedra with no interiors, like the heptahedron.
Eventually, in Chapter 5, we learn that V-E+F=2. Cromwell gives us a very well informed discussion of what Descartes knew of the so-called "Euler Formula," and leads us to the conclusion that, though Descartes was close, none the less we are correct to attribute the formula to Euler rather than to Descartes.
Next, we learn about flexibility and rigidity. Cromwell tells the story of Robert Connelly and his search for a flexible polyhedron. At a topology conference in 1975, after several years working on the problem, he heard the sickening rumor that someone had just discovered a flexible polyhedron. The story has a happy ending, though. He hunted down the source of the rumor, and found out that it was a rumor about himself. Two years later, the rumor came true, and he found his flexible polyhedron.
There's more: stellations, symmetry groups, non-convex vertex-transitive polyhedra, coloring problems, compound polyhedra. The hundreds of illustrations are clear and informative. Most readers will be unable to resist the author's suggestion that they make physical models to understand better the shapes in the illustrations.
A small production error should be noted. The text at the bottom of page 309 is repeated at the top of page 310. This seems to happen all too often when manuscripts are prepared electronically.
Polyhedra is a wonderful book, worthy of many readings. It shows us, and others, how we find beauty and mystery in mathematics. It belongs in every university library, and on most mathematicians' book shelves.
Ed Sandifer (firstname.lastname@example.org) is a professor of mathematics at Western Connecticut State University, Contributed Papers Coordinator for the Northeastern Section of the MAA and an avid runner.