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

Princeton University Press (2008)

ISBN13: 978-0-691-12677-7

Leonhard Euler’s polyhedron formula describes the structure of many objects—from soccer balls and gemstones to Buckminster Fuller’s buildings and giant all-carbon molecules. Yet Euler’s formula is so simple it can be explained to a child. *Euler’s Gem* tells the illuminating story of this indispensable mathematical idea.

From ancient Greek geometry to today’s cutting-edge research, *Euler’s Gem* celebrates the discovery of Euler’s beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of *V* vertices, *E* edges, and *F* faces satisfies the equation *V*–*E*+*F*=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula’s scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler’s formula. Using wonderful examples and numerous illustrations, Richeson presents the formula’s many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map.

Filled with a who’s who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem’s development, *Euler’s Gem* will fascinate every mathematics enthusiast.

**Sample chapter:** Introduction (html, pdf)

## Awards and Honors

*Euler’s Gem* was the recipient of the ** 2010 Euler Book Prize**, by the Mathematical Association of America. From the MAA website: “The Euler Book Prize shall be given to the author or authors of an outstanding book about mathematics. Mathematical monographs at the undergraduate level, histories, biographies, works of mathematical fiction, and anthologies shall be among those types of books eligible for the Prize. They shall be judged on clarity of exposition and the degree to which they have had or show promise of having a positive impact on the public’s view of mathematics in the United States and Canada.” You can find photos of the Joint Prize Session on the AMS website and in MAA Focus, and you can read the prize citation (pdf).

*Euler’s Gem* was selected by *Choice Magazine* as an* Outstanding Academic Title* for 2009.

*Euler’s Gem* was the book of the month selection for the *Scientific American Book Club* in February 2009.

## Reviews

“David Richeson’s *Euler’s Gem* does an outstanding job of explaining serious mathematics to a general audience, and I plan to recommend it to the next stranger I meet on an airplane.”—Jeremy L. Martin, *Notices of the AMS*

“The author has achieved a remarkable feat, introducing a naïve reader to a rich history without compromising the insights and without leaving out a delicious detail. Furthermore, he describes the development of topology from a suggestion by Gottfried Leibniz to its algebraic formulation by Emmy Noether, relating all to Euler’s formula. This book will be valuable to every library with patrons looking for an awe-inspiring experience.”—J. McCleary, *Choice Magazine*

“I highly recommend this book for teachers interested in geometry or topology, particularly for university faculty. The examples, proofs, and historical anecdotes are interesting, informative, and useful for encouraging classroom discussions. Advanced students will also glimpse the broad horizons of mathematics by reading (and working through) the book.”—Dustin L. Jones, *Mathematics Teacher*

“This is an excellent book about a great man and a timeless formula. Well within the reach of the intelligent layperson, it is also a good book to use as a resource for a course where the students are required to make presentations.”—Charles Ashbacher, *Journal of Recreational Mathematics*

“The book contains hundreds of diagrams, all beautifully rendered and seamlessly integrated into the narrative. Even without burdensome calculations, there are places where Richeson demands much from his readers, but he rewards them with a fast-paced and wide-ranging survey of some very fascinating facets of modern mathematics.”—Robert Bradley, *Times Higher Education*

“I liked Richeson’s style of writing. He is enthusiastic and humorous. It was a pleasure reading this book, and I recommend it to everyone who is not afraid of mathematical arguments and has ever wondered what this field of ‘rubber-sheet geometry’ is about. You will not be disappointed.”—Jeanine Daems, *The Mathematical Intelligencer*

“It’s an excellent account of the historical context of today’s topology and combinatorics.”—Tony Mann,*Times Higher Education*

## Endorsements

“*Euler’s Gem* is a thoroughly satisfying meditation on one of mathematics’ loveliest formulas. The author begins with Euler’s act of seeing what no one previously had, and returns repeatedly to the resulting formula with ever more careful emendations and ever-widening points of view. This highly nuanced narrative sweeps the reader into the cascade of interlocking ideas which undergird modern topology and lend it its power and beauty.”—Donal O’Shea, author of *The Poincaré Conjecture: In Search of the Shape of the Universe*

“Beginning with Euler’s famous polyhedron formula, continuing to modern concepts of ‘rubber geometry,’ and advancing all the way to the proof of Poincaré’s Conjecture, Richeson’s well-written and well-illustrated book is a gentle tour de force of topology.”—George G. Szpiro, author of *Poincaré’s Prize: The Hundred-Year Quest to Solve One of Math’s Greatest Puzzles*

“A fascinating and accessible excursion through two thousand years of mathematics. From Plato’s Academy, via the bridges of Königsberg, to the world of knots, soccer balls, and geodesic domes, the author’s enthusiasm shines through. This attractive introduction to the origins of topology deserves to be widely read.”—Robin Wilson, author of *Four Colors Suffice: How the Map Problem Was Solved*

“Appealing and accessible to a general audience, this well-organized, well-supported, and well-written book contains vast amounts of information not found elsewhere. *Euler’s Gem* is a significant and timely contribution to the field.”—Edward Sandifer, Western Connecticut State University

“*Euler’s Gem* is a very good book. It succeeds in explaining complicated concepts in engaging layman’s terms. Richeson is keenly aware of where the difficult twists and turns are located, and he covers them to satisfaction. This book is engaging and a joy to read.”—Alejandro López-Ortiz, University of Waterloo

## Translations

- Japanese
- Portuguese (forthcoming)
- Korean (forthcoming)
- Chinese (forthcoming)

## Table of Contents

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