The book is organized historically, and reviewer Robert Bradley divides the topics of the book into three parts. The first part discusses the earlier history of polyhedra, including the works of
Pythagoras,
Thales,
Euclid, and
Johannes Kepler, and the discovery by
René Descartes of a polyhedral version of the
Gauss–Bonnet theorem (later seen to be equivalent to Euler's formula). It surveys the life of
Euler, his discovery in the early 1750s that the
Euler characteristic V-E+F (the number of vertices minus the number of edges plus the number of faces) is equal to 2 for all
convex polyhedra, and his flawed attempts at a proof, and concludes with the first rigorous proof of this identity in 1794 by
Adrien-Marie Legendre, based on
Girard's theorem relating the angular excess of triangles in
spherical trigonometry to their area. Although polyhedra are geometric objects, ''Euler's Gem'' argues that Euler discovered his formula by being the first to view them topologically (as abstract incidence patterns of vertices, faces, and edges), rather than through their geometric distances and angles. (However, this argument is undermined by the book's discussion of similar ideas in the earlier works of Kepler and Descartes.) The birth of topology is conventionally marked by an earlier contribution of Euler, his 1736 work on the
Seven Bridges of Königsberg, and the middle part of the book connects these two works through the
theory of graphs. It proves Euler's formula in a topological rather than geometric form, for
planar graphs, and discusses its uses in proving that these graphs have vertices of low
degree, a key component in proofs of the
four color theorem. It even makes connections to
combinatorial game theory through the graph-based games of
Sprouts and Brussels Sprouts and their analysis using Euler's formula. In the third part of the book, Bradley moves on from the topology of the plane and the sphere to arbitrary topological surfaces. For any surface, the Euler characteristics of all subdivisions of the surface are equal, but they depend on the surface rather than always being 2. Here, the book describes the work of
Bernhard Riemann,
Max Dehn, and
Poul Heegaard on the
classification of manifolds, in which it was shown that the two-dimensional
compact topological surfaces can be completely described by their Euler characteristics and their
orientability. Other topics discussed in this part include
knot theory and the Euler characteristic of
Seifert surfaces, the
Poincaré–Hopf theorem, the
Brouwer fixed point theorem,
Betti numbers, and
Grigori Perelman's proof of the
Poincaré conjecture. An appendix includes instructions for creating paper and soap-bubble models of some of the examples from the book. ==Audience and reception==