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This title in other editionsEuler's Gem: The Polyhedron Formula and the Birth of Topologyby David S. Richeson
Synopses & ReviewsPublisher Comments:"Euler's Gem is a thoroughly satisfying meditation on one of mathematics' loveliest formulas. The author begins with Euler's act of seeing what no one previously had, and returns repeatedly to the resulting formula with ever more careful emendations and everwidening points of view. This highly nuanced narrative sweeps the reader into the cascade of interlocking ideas which undergird modern topology and lend it its power and beauty."Donal O'Shea, author of The Poincaré Conjecture: In Search of the Shape of the Universe
"Beginning with Euler's famous polyhedron formula, continuing to modern concepts of 'rubber geometry,' and advancing all the way to the proof of Poincaré's Conjecture, Richeson's wellwritten and wellillustrated book is a gentle tour de force of topology."George G. Szpiro, author of Poincaré's Prize: The HundredYear Quest to Solve One of Math's Greatest Puzzles
"A fascinating and accessible excursion through two thousand years of mathematics. From Plato's Academy, via the bridges of Königsberg, to the world of knots, soccer balls, and geodesic domes, the author's enthusiasm shines through. This attractive introduction to the origins of topology deserves to be widely read."Robin Wilson, author of Four Colors Suffice: How the Map Problem Was Solved
"Appealing and accessible to a general audience, this wellorganized, wellsupported, and wellwritten book contains vast amounts of information not found elsewhere. Euler's Gem is a significant and timely contribution to the field."Edward Sandifer, Western Connecticut State University
"Euler's Gem is a very good book. It succeeds in explaining complicated concepts in engaging layman's terms. Richeson is keenly aware of where the difficult twists and turns are located, and he covers them to satisfaction. This book is engaging and a joy to read."Alejandro LópezOrtiz, University of Waterloo Synopsis:"Euler's Gem is a thoroughly satisfying meditation on one of mathematics' loveliest formulas. The author begins with Euler's act of seeing what no one previously had, and returns repeatedly to the resulting formula with ever more careful emendations and everwidening points of view. This highly nuanced narrative sweeps the reader into the cascade of interlocking ideas which undergird modern topology and lend it its power and beauty."Donal O'Shea, author of The Poincaré Conjecture: In Search of the Shape of the Universe
"Beginning with Euler's famous polyhedron formula, continuing to modern concepts of 'rubber geometry,' and advancing all the way to the proof of Poincaré's Conjecture, Richeson's wellwritten and wellillustrated book is a gentle tour de force of topology."George G. Szpiro, author of Poincaré's Prize: The HundredYear Quest to Solve One of Math's Greatest Puzzles "A fascinating and accessible excursion through two thousand years of mathematics. From Plato's Academy, via the bridges of Königsberg, to the world of knots, soccer balls, and geodesic domes, the author's enthusiasm shines through. This attractive introduction to the origins of topology deserves to be widely read."Robin Wilson, author of Four Colors Suffice: How the Map Problem Was Solved "Appealing and accessible to a general audience, this wellorganized, wellsupported, and wellwritten book contains vast amounts of information not found elsewhere. Euler's Gem is a significant and timely contribution to the field."Edward Sandifer, Western Connecticut State University "Euler's Gem is a very good book. It succeeds in explaining complicated concepts in engaging layman's terms. Richeson is keenly aware of where the difficult twists and turns are located, and he covers them to satisfaction. This book is engaging and a joy to read."Alejandro LópezOrtiz, University of Waterloo Synopsis:Leonhard Euler's polyhedron formula describes the structure of many objectsfrom soccer balls and gemstones to Buckminster Fuller's buildings and giant allcarbon 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 cuttingedge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its farreaching 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 VE+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenthcentury 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 twentiethcentury 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. About the AuthorDavid S. Richeson is associate professor of mathematics at Dickinson College.
Table of ContentsPreface 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: Combing 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: Henri Poincaré and the Ascendance of Topology 253 Epilogue The MillionDollar Question 265 Acknowledgements 271 Appendix A Build Your Own Polyhedra and Surfaces 273 Appendix B Recommended Readings 283 Notes 287 References 295 Illustration Credits 309 Index 311 What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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