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

ISBN-10: 0691154570
ISBN-13: 9780691154572
Edition: 2008
List price: $14.95 Buy it from $9.25
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Description: 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  More...

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Book details

List price: $14.95
Copyright year: 2008
Publisher: Princeton University Press
Publication date: 4/20/2012
Binding: Paperback
Pages: 336
Size: 6.25" wide x 9.25" long x 1.00" tall
Weight: 1.210
Language: English

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 ofVvertices,Eedges, andFfaces satisfies the equationV-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.

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

Preface
Introduction
Leonhard Euler and His Three "Great" Friends
What Is a Polyhedron?
The Five Perfect Bodies
The Pythagorean Brotherhood and Plato's Atomic Theory
Euclid and His Elements
Kepler's Polyhedral Universe
Euler's Gem
Platonic Solids, Golf Balls, Fullerenes, and Geodesic Domes
Scooped by Descartes?
Legendre Gets It Right
A Stroll through K�nigsberg
Cauchy's Flattened Polyhedra
Planar Graphs, Geoboards, and Brussels Sprouts
It's a Colorful World
New Problems and New Proofs
Rubber Sheets, Hollow Doughnuts, and Crazy Bottles
Are They the Same, or Axe They Different?
A Knotty Problem
Combing the Hair on a Coconut
When Topology Controls Geometry
The Topology of Curvy Surfaces
Navigating in n Dimensions
Henri Poincar� and the Ascendance of Topology
Epilogue: The Million-Dollar Question
Acknowledgments
Build Your Own Polyhedra and Surfaces
Recommended Readings
Notes
References
Illustration Credits
Index

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