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Euler's Gem: The Polyhedron Formula and the Birth of Topology

David S. Richeson
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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“Euler’s gem” of the title is the famous formula for polyhedra. The author, David Richeson, calls his book “a history and celebration of topology.” He uses the Euler formula as the centerpiece of a story that goes back to topology’s prehistory with the Greeks, then moves though the Renaissance and forward into the critical developments of the eighteenth and nineteenth centuries. Roughly the last quarter of the book is devoted to the birth of modern topology in the twentieth century.

Euler’s Gem has no formal prerequisites. High school mathematics — algebra, geometry and a bit of trigonometry — are said to be sufficient. The author views his potential readership as self-selecting: if someone wants to read it, he or she should be capable of working their way through it. This is not entirely plausible because there are several sections later in the book — for example, a discussion of the Euler characteristic in n dimensions — that would be an adequate summary for mathematically trained readers, but wouldn’t mean much to a neophyte.

This is not a textbook and there are no exercises. Nonetheless, the author does offer some proofs, although he is careful not to make them look too formal. The early chapters of this book focus on regular polyhedra and discuss what the author calls the pre-Eulerian view of polyhedra. The author considers the contributions of the Greeks and expresses surprise that they never discovered Euler’s formula. Descartes’ contributions and the controversy about who got to the famous formula first are discussed at some length. Although this would seem to be a detour from the main story, it does support one of the author’s objectives by showing that mathematics develops in a highly nonlinear fashion with a mixture of incremental progress, big leaps, confusions, corrections and rivalries.

After discussing Euler’s life, his work and efforts leading up to his formula, the author describes contributions from Descartes, Legendre and Cauchy, and provides some simple applications. Successive chapters take up graph theory, surfaces and their relationship to Euler’s formula, knot theory, dynamical systems, and geometry. The range of topics is considerable, but the result is not quite cohesive.

The parts of the book that work best are places where the author skillfully takes the reader through an argument. There is a nice example where he proves Pick’s theorem on the area of a polygon whose vertices are lattice points. In another place he shows that there is a winning strategy for Conway’s Brussels sprouts game. Both arguments rely on Euler’s formula and emphasize its versatility. The book would benefit from fewer topics and more emphasis on examples like these.

This book does have an excellent bibliography and a very useful appendix with recommended reading.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.


Preface ix
Introduction 1
Chapter 1: Leonhard Euler and His Three "Great" Friends 10
Chapter 2: What Is a Polyhedron? 27
Chapter 3: The Five Perfect Bodies 31
Chapter 4: The Pythagorean Brotherhood and Plato's Atomic Theory 36
Chapter 5: Euclid and His Elements 44
Chapter 6: Kepler's Polyhedral Universe 51
Chapter 7: Euler's Gem 63
Chapter 8: Platonic Solids, Golf Balls, Fullerenes, and Geodesic Domes 75
Chapter 9: Scooped by Descartes? 81
Chapter 10: Legendre Gets It Right 87
Chapter 11: A Stroll through Königsberg 100
Chapter 12: Cauchy's Flattened Polyhedra 112
Chapter 13: Planar Graphs, Geoboards, and Brussels Sprouts 119
Chapter 14: It's a Colorful World 130
Chapter 15: New Problems and New Proofs 145
Chapter 16: Rubber Sheets, Hollow Doughnuts, and Crazy Bottles 156
Chapter 17: Are They the Same, or Are They Different? 173
Chapter 18: A Knotty Problem 186
Chapter 19: Combing the Hair on a Coconut 202
Chapter 20: When Topology Controls Geometry 219
Chapter 21: The Topology of Curvy Surfaces 231
Chapter 22: Navigating in n Dimensions 241
Chapter 23: Henri Poincaré and the Ascendance of Topology 253
Epilogue The Million-Dollar Question 265
Acknowledgements 271
Appendix A Build Your Own Polyhedra and Surfaces 273
Appendix B Recommended Readings 283
Notes 287
References 295
Illustration Credits 309
Index 311