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

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

by David S. Richeson

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Overview

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.

Product Details

ISBN-13: 9780691154572
Publisher: Princeton University Press
Publication date: 04/15/2012
Edition description: Reprint
Pages: 336
Product dimensions: 6.00(w) x 9.10(h) x 0.90(d)

About the Author

David S. Richeson is associate professor of mathematics at Dickinson College.

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 Künigsberg 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

What People are Saying About This

Donal O'Shea

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 Poincare Conjecture: In Search of the Shape of the Universe"

Robin Wilson

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"

Edward Sandifer

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

Alejandro Lopez-Ortiz

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 Lopez-Ortiz, University of Waterloo

From the Publisher

“David Richeson’s Euler’s Gem does an outstanding job of explaining serious mathematics to a general audience. . . . The book succeeds at showing the reader a lot of attractive mathematics with a well-chosen level of technical detail.”—Jeremy L. Martin, Notices of the AMS

Szpiro

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 "Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles"

Customer Reviews