Euler's Gem The Polyhedron Formula and the Birth of Topology~tqw~ dark

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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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