21. The Golden Ratio

Chapters in this book

1. Frontmatter i
2. Contents vii
3. Prefaces. Art through Mathematical Eyes ix
4. 1. Thales of Miletus 1
5. 2. Triangles of Equal Area 3
6. 3. Quadrilaterals 6
7. 4. Perfect Numbers and Triangular Numbers 9
8. 5. The Pythagorean Theorem I 13
9. 6. The Pythagorean Theorem II 16
10. 7. Pythagorean Triples 20
11. 8. The Square Root of 2 23
12. 9. A Repertoire of Means 26
13. 10. More about Means 29
14. 11. Two Theorems from Euclid 32
15. 12. Different, yet the Same 36
16. 13. One Theorem, Three Proofs 39
17. 14. The Prime Numbers 42
18. 15. Two Prime Mysteries 45
19. 16. 0.999. . . = ? 49
20. 17. Eleven 53
21. 18. Euclidean Constructions 56
22. 19. Hexagons 59
23. 20. Fibonacci Numbers 62
24. 21. The Golden Ratio 66
25. 22. The Pentagon 70
26. 23. The 17-Sided Regular Polygon 73
27. 24. Fifty 77
28. 25. Doubling the Cube 81
29. 26. Squaring the Circle 84
30. 27. Archimedes Measures the Circle 88
31. 28. The Digit Hunters 91
32. 29. Conics 94
33. 30. 3/3=4/4 99
34. 31. The Harmonic Series 102
35. 32. Ceva’s Theorem 105
36. 33. e 108
37. 34. Spira Mirabilis 112
38. 35. The Cycloid 116
39. 36. Epicycloids and Hypocycloids 119
40. 37. The Euler Line 123
41. 38. Inversion 126
42. 39. Steiner’s Porism 130
43. 40. Line Designs 134
44. 41. The French Connection 138
45. 42. The Audible Made Visible 141
46. 43. Lissajous Figures 143
47. 44. Symmetry I 146
48. 45. Symmetry II 149
49. 46. The Reuleaux Triangle 154
50. 47. Pick’s Theorem 157
51. 48. Morley’s Theorem 160
52. 49. The Snowflake Curve 164
53. 50. Sierpinski’s Triangle 167
54. 51. Beyond Infinity 170
55. Appendix: Proofs of Selected Theorems Mentioned in This Book 175
56. Bibliography 183
57. Index 185