Curves for the Mathematically Curious is a thoughtfully curated collection of ten mathematical curves, selected by Julian Havil for their significance, mathematical interest, and beauty. Each chapter gives an account of the history and definition of a curve, providing a glimpse into the elegant and often surprising mathematics involved in its creation and evolution. In telling the ten stories, Havil introduces many mathematicians and other innovators, some whose fame has withstood the passing of years and others who have slipped into comparative obscurity. You will meet Pierre Bézier, who is known for his ubiquitous and eponymous curves, and Adolphe Quetelet, who trumpeted the ubiquity of the normal curve but whose name now hides behind the modern body mass index. These and other ingenious thinkers engaged with the challenges, incongruities, and insights to be found in these remarkable curves―and now you can share in this adventure.
Author(s): Julian Havil
Publisher: PRINCETON UNIVERSITY PRESS
Year: 2019
Language: English
Pages: 281
Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 10
Preface......Page 14
Acknowledgments......Page 18
1.1 An Unusual Parametrization…......Page 22
1.2 …Yet a Natural Parametrization......Page 25
1.3 A Challenge......Page 28
1.4 One Curve, Many Names......Page 32
2.1 Naive Thoughts......Page 36
2.2 Profound Thoughts......Page 38
2.3 Differentiability......Page 41
2.4 Weierstrass’s Proof......Page 43
2.5 The Aftermath......Page 47
2.6 Final Thoughts......Page 50
3.1 Bézier’s Curve of Curves......Page 53
3.2 Bézier and Bernstein......Page 59
3.3 Bézier and Casteljau......Page 64
3.4 The Story of Lump......Page 66
3.5 The Story of O......Page 67
4.1 Old Logarithms......Page 71
4.2 A Thorny Problem......Page 74
4.3 Computation......Page 81
4.4 New Logarithms......Page 83
5.1 Problems of Antiquity......Page 87
5.2 Some Greek Constructions......Page 89
5.3 The Quadratrix and Trisection......Page 96
5.4 The Quadratrix and Circle Squaring......Page 99
6.1 Je Le Vois, Mais Je Ne Le Crois Pas......Page 106
6.2 Peano’s Function......Page 111
6.3 Hilbert’s Curve......Page 116
6.4 Peano’s Curve......Page 122
CHAPTER SEVEN Curves of Constant Width......Page 125
7.1 The Reuleaux Triangle …......Page 126
7.2 …And Its Generalizations......Page 131
7.3 And Their Generalization …......Page 138
7.4 A Circle in All but Name?......Page 144
8.1 A Fruitful Question......Page 147
8.2 An Answer but Not a Solution......Page 149
8.3 Approximating the Impossible......Page 152
8.4 Error Curves......Page 159
8.5 The Error Curve......Page 165
8.6 The Normal Distribution......Page 172
9.1 A Matter of Symmetry......Page 176
9.2 Historical Errors......Page 178
9.3 The Curve Identified......Page 183
9.4 Hyperbolic Functions......Page 190
9.5 The Chain Inverted......Page 194
9.6 A Bumpy Road......Page 200
CHAPTER TEN Elliptic Curves......Page 204
10.1 Elliptic Ambiguity......Page 205
10.2 Problems, Problems, Problems......Page 210
10.3 Common Ground......Page 214
10.4 The Congruent Number Problem......Page 217
10.5 An Arithmetic......Page 224
10.6 Fertile Fields......Page 230
10.7 Cryptography......Page 241
10.8 Apologia......Page 246
Perhaps the Most Important Curve of All......Page 248
Appendix A The Title Page......Page 250
Appendix B Conics Encapsulated......Page 253
Appendix C A Trigonometric Variant for the Bézier Curve......Page 255
Appendix D Envelopes......Page 257
Appendix E The Mathematics of an Arch......Page 261
Appendix F The Simple Pendulum......Page 263
Appendix G Fibonacci’s Method......Page 264
References......Page 268
Index......Page 276
