Euler's Gem

by Richeson, David S.

• ISBN13: 9780691154572
• ISBN10: 0691154570
• Edition: Reprint
• Format: Paperback
• Copyright: 2012-03-26
• Publisher: Princeton Univ Pr

Summary

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 Gemtells 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 Vvertices, Eedges, and Ffaces 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 Gemwill fascinate every mathematics enthusiast.

Author Biography

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

Table of Contents

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