Yearning for the impossible: the surprising truths of mathematics [Second edition] 9781138596214, 9781138586109, 1138596213, 1138586102

Table of contents :
Cover......Page 1
Half title......Page 2
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Preface to the Second Edition......Page 8
Preface......Page 10
Contents......Page 14
Chapter1 The Irrational......Page 18
1.1 The Pythagorean Dream......Page 19
1.2 The Pythagorean Theorem......Page 23
1.3 Irrational Triangles......Page 28
1.4 The Pythagorean Nightmare......Page 32
1.5 Explaining the Irrational......Page 35
1.6 The Continued Fraction for 2......Page 39
1.7 Equal Temperament......Page 45
Chapter2 The Imaginary......Page 50
2.1 Negative Numbers......Page 51
2.2 Imaginary Numbers......Page 55
2.3 Solving Cubic Equations......Page 58
2.4 Real Solutions via Imaginary Numbers......Page 61
2.5 WhereWere Imaginary Numbers before 1572?......Page 62
2.6 Geometry ofMultiplication......Page 66
2.7 Complex Numbers GiveMore thanWe Asked for......Page 72
2.8 Why Call Them “Complex” Numbers?......Page 76
Chapter3 The Horizon......Page 79
3.1 Parallel Lines......Page 81
3.2 Coordinates......Page 84
3.3 Parallel Lines and Vision......Page 88
3.4 Drawing withoutMeasurement......Page 94
3.5 The Theorems of Pappus and Desargues......Page 100
3.6 The Little Desargues Theorem......Page 105
3.7 What Are the Laws of Algebra?......Page 109
3.8 Projective Addition andMultiplication......Page 116
Chapter4 The Infinitesimal......Page 121
4.1 Length and Area......Page 122
4.2 Volume......Page 124
4.3 Volume of a Tetrahedron......Page 126
4.4 The Circle......Page 130
4.5 The Parabola......Page 133
4.6 The Slopes of Other Curves......Page 137
4.7 Slope and Area......Page 141
4.8 The Value of ¼......Page 145
4.9 Ghosts of Departed Quantities......Page 147
Chapter5 Curved Space......Page 151
5.1 Flat Space andMedieval Space......Page 152
5.2 The 2-Sphere and the 3-Sphere......Page 155
5.3 Flat Surfaces and the Parallel Axiom......Page 162
5.4 The Sphere and the Parallel Axiom......Page 165
5.5 Non-Euclidean Geometry......Page 170
5.6 Negative Curvature......Page 174
5.7 The Hyperbolic Plane......Page 177
5.8 Hyperbolic Space......Page 184
5.9 Mathematical Space and Actual Space......Page 185
Chapter6 The Fourth Dimension......Page 190
6.1 Arithmetic of Pairs......Page 191
6.2 Searching for an Arithmetic of Triples......Page 193
6.3 Why n-tuples Are Unlike Numbers when n ¸ 3......Page 195
6.4 Quaternions......Page 199
6.5 The Four-Square Theorem......Page 204
6.6 Quaternions and Space Rotations......Page 207
6.7 Symmetry in Three Dimensions......Page 211
6.8 Tetrahedral Symmetry and the 24-Cell......Page 215
6.9 The Regular Polytopes......Page 219
Chapter7 The Ideal......Page 224
7.1 Discovery and Invention......Page 225
7.2 Division with Remainder......Page 228
7.3 The Euclidean Algorithm......Page 232
7.4 Unique Prime Factorization......Page 235
7.5 Gaussian Integers......Page 237
7.6 Gaussian Primes......Page 241
7.7 Rational Slopes and Rational Angles......Page 244
7.8 Unique Prime Factorization Lost......Page 246
7.9 Ideals—Unique Prime Factorization Regained......Page 251
Chapter8 Periodic Space......Page 256
8.1 The Impossible Tribar......Page 258
8.2 The Cylinder and the Plane......Page 260
8.3 Where theWild Things Are......Page 263
8.4 PeriodicWorlds......Page 266
8.5 Periodicity and Topology......Page 270
8.6 A Brief History of Periodicity......Page 275
8.7 Non-Euclidean Periodicity......Page 280
Chapter9 The Infinite......Page 285
9.1 Finite and Infinite......Page 286
9.2 Potential and Actual Infinity......Page 288
9.3 The Uncountable......Page 291
9.4 The Diagonal Argument......Page 294
9.5 The Transcendental......Page 298
9.6 Yearning for Completeness......Page 303
Epilogue......Page 306
References......Page 310
Index......Page 314