Euler's Gem : The Polyhedron Formula and the Birth of Topology / David S. Richeson.

Format:

Book

Author/Creator:

Richeson, David S., author.

Language:

English

Subjects (All):

Polyhedra.

Topology--History.

Topology.

Physical Description:

1 online resource (332 p.)

Edition:

Course Book

Place of Publication:

Princeton, NJ : Princeton University Press, [2012]

Language Note:

English

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 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.

Contents:

Front matter

Contents

Preface

Introduction

Chapter 1. Leonhard Euler and His Three "Great" Friends

Chapter 2. What Is a Polyhedron?

Chapter 3. The Five Perfect Bodies

Chapter 4. The Pythagorean Brotherhood and Plato's Atomic Theory

Chapter 5. Euclid and His "Elements"

Chapter 6. Kepler's Polyhedral Universe

Chapter 7. Euler's Gem

Chapter 8. Platonic Solids, Golf Balls, Fullerenes, and Geodesic Domes

Chapter 9. Scooped by Descartes?

Chapter 10. Legendre Gets It Right

Chapter 11. A Stroll through Königsberg

Chapter 12. Cauchy's Flattened Polyhedra

Chapter 13. Planar Graphs, Geoboards, and Brussels Sprouts

Chapter 14. It's a Colorful World

Chapter 15. New Problems and New Proofs

Chapter 16. Rubber Sheets, Hollow Doughnuts, and Crazy Bottles

Chapter 17. Are They the Same, or Are They Different?

Chapter 18. A Knotty Problem

Chapter 19. Combing the Hair on a Coconut

Chapter 20. When Topology Controls Geometry

Chapter 21. The Topology of Curvy Surfaces

Chapter 22. Navigating in n Dimensions

Chapter 23. Henri Poincaré and the Ascendance of Topology

Epilogue: The Million-Dollar Question

Acknowledgments

Appendix A. Build Your Own Polyhedra and Surfaces

Appendix B. Recommended Readings

Notes

References

Illustration Credits

Index

Notes:

Description based upon print version of record.

Includes bibliographical references (p. [295]-308) and index.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)

ISBN:

1-283-12926-4

9786613129260

1-4008-3856-8

OCLC:

753980256