Chapter 20. When Topology Controls Geometry

Chapters in this book

1. Frontmatter i
2. Contents vii
3. Preface to the Princeton Science Library Edition ix
4. Preface xiii
5. Introduction 1
6. Chapter 1. Leonhard Euler and His Three “Great” Friends 10
7. Chapter 2. What Is a Polyhedron? 27
8. Chapter 3. The Five Perfect Bodies 31
9. Chapter 4. The Pythagorean Brotherhood and Plato’s Atomic Theory 36
10. Chapter 5. Euclid and His Elements 44
11. Chapter 6. Kepler’s Polyhedral Universe 51
12. Chapter 7. Euler’s Gem 63
13. Chapter 8. Platonic Solids, Golf Balls, Fullerenes, and Geodesic Domes 75
14. Chapter 9. Scooped by Descartes? 81
15. Chapter 10. Legendre Gets It Right 87
16. Chapter 11. A Stroll through Königsberg 100
17. Chapter 12. Cauchy’s Flattened Polyhedra 112
18. Chapter 13. Planar Graphs, Geoboards, and Brussels Sprouts 119
19. Chapter 14. It’s a Colorful World 130
20. Chapter 15. New Problems and New Proofs 145
21. Chapter 16. Rubber Sheets, Hollow Doughnuts, and Crazy Bottles 156
22. Chapter 17. Are They the Same, or Are They Different? 173
23. Chapter 18. A Knotty Problem 186
24. Chapter 19. Combing the Hair on a Coconut 202
25. Chapter 20. When Topology Controls Geometry 219
26. Chapter 21. The Topology of Curvy Surfaces 231
27. Chapter 22. Navigating in n Dimensions 241
28. Chapter 23. Henri Poincaré and the Ascendance of Topology 253
29. Epilogue: The Million-Dollar Question 265
30. Acknowledgments 271
31. Appendix A: Build Your Own Polyhedra and Surfaces 273
32. Appendix B: Recommended Readings 283
33. Notes 287
34. References 295
35. Illustration Credits 309
36. Index 311