Euler's Gem The Polyhedron Formula and the Birth of Topology
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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 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.
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All the information you need in one place! Each Study Brief is a summary of one specific subject; facts, figures, and explanations to help you learn faster.
List price: $14.95
Copyright year: 2012
Publisher: Princeton University Press
Publication date: 4/15/2012
Size: 6.25" wide x 9.25" long x 1.00" tall
David S. Richeson is associate professor of mathematics at Dickinson College.
|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|
|Platonic Solids, Golf Balls, Fullerenes, and Geodesic Domes|
|Scooped by Descartes?|
|Legendre Gets It Right|
|A Stroll through Kï¿½nigsberg|
|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 Poincarï¿½ and the Ascendance of Topology|
|Epilogue: The Million-Dollar Question|
|Build Your Own Polyhedra and Surfaces|