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

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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.


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 Combining 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 Hermi 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

Index 309


Product Details


    * ISBN: 0691126771

    * ISBN-13: 9780691126777

    * Format: Hardcover, 336pp

    * Publisher: Princeton University Press

    * Pub. Date: September 25, 2008


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