Maor, E: Beautiful Geometry

Prefaces ix 1. Thales of Miletus 1 2. Triangles of Equal Area 3 3. Quadrilaterals 6 4. Perfect Numbers and Triangular Numbers 9 5. The Pythagorean Theorem I 13 6. The Pythagorean Theorem II 16 7. Pythagorean Triples 20 8. The Square Root of 2 23 9. A Repertoire of Means 26 10. More about Means 29 11. Two Theorems from Euclid 32 12. Different, yet the Same 36 13. One Theorem, Three Proofs 39 14. The Prime Numbers 42 15. Two Prime Mysteries 45 16. 0.999... = ? 49 17. Eleven 53 18. Euclidean Constructions 56 19. Hexagons 59 20. Fibonacci Numbers 62 21. The Golden Ratio 66 22. The Pentagon 70 23. The 17-Sided Regular Polygon 73 24. Fifty 77 25. Doubling the Cube 81 26. Squaring the Circle 84 27. Archimedes Measures the Circle 88 28. The Digit Hunters 91 29. Conics 94 30. 3/4 = 4/4 99 31. The Harmonic Series 102 32. Ceva's Theorem 105 33. e 108 34. Spira Mirabilis 112 35. The Cycloid 116 36. Epicycloids and Hypocycloids 119 37. The Euler Line 123 38. Inversion 126 39. Steiner's Porism 130 40. Line Designs 134 41. The French Connection 138 42. The Audible Made Visible 141 43. Lissajous Figures 143 44. Symmetry I 146 45. Symmetry II 149 46. The Reuleaux Triangle 154 47. Pick's Theorem 157 48. Morley's Theorem 160 49. The Snowflake Curve 164 50. Sierpinski's Triangle 167 51. Beyond Infinity 170 APPENDIX: Proofs of Selected Theorems Mentioned in This Book 175 Quadrilaterals 175 Pythagorean Triples 176 A Proof That 2 Is Irrational 176 Euclid's Proof of the Infinitude of the Primes 176 The Sum of a Geometric Progression 177 The Sum of the First n Fibonacci Numbers 177 Construction of a Regular Pentagon 177 Ceva's Theorem 179 Some Properties of Inversion 180 Bibliography 183 Index 185

Eli Maor is the author of To Infinity and Beyond, e: The Story of a Number, Trigonometric Delights, Venus in Transit, and The Pythagorean Theorem: A 4,000-Year History (all Princeton), and has taught the history of mathematics at Loyola University Chicago. Eugen Jost is a Swiss artist whose work is strongly influenced by mathematics.

"Reading Beautiful Geometry is like touring a personal art collection, except the masterpieces here are elegantly presented theorems and constructions by ancient and modern mathematicians. Eli Maor teaches the thought process of a geometry connoisseur as he highlights important details in each of these gems, and Eugen Jost illustrates these ideas with colorful and creative artworks. Readers who like to see real math in their math books will enjoy how Beautiful Geometry brings these enduring mathematical achievements to life."--George Hart, mathematical sculptor