Curves for the Mathematically Curious: An Anthology of the Unpredictable, Historical, Beautiful, and Romantic

by Julian Havil

Overview

Ten amazing curves personally selected by one of today's most important math writers

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 one 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.

Curves for the Mathematically Curious is a rigorous and enriching mathematical experience for anyone interested in curves, and the book is designed so that readers who choose can follow the details with pencil and paper. Every curve has a story worth telling.

Product Details

ISBN-13:	9780691206134
Publisher:	Princeton University Press
Publication date:	11/02/2021
Pages:	280
Product dimensions:	5.25(w) x 8.00(h) x (d)

About the Author

Julian Havil is the author of John Napier: Life, Logarithms, and Legacy; The Irrationals: A Story of the Numbers You Can't Count On; Impossible? Surprising Solutions to Counterintuitive Conundrums; Nonplussed! Mathematical Proof of Implausible Ideas; and Gamma: Exploring Euler's Constant (all Princeton). He is a retired former master at Winchester College, England, where he taught mathematics for more than three decades.

Table of Contents

Preface xiii

Acknowledgments xvii

Chapter 1 The Euler Spiral 1

1.1 An Unusual Parametrization… 1

1.2 …Yet a Natural Parametrization 4

1.3 A Challenge 7

1.4 One Curve, Many Names 11

Chapter 2 The Weierstrass Curve 15

2.1 Naive Thoughts 15

2.2 Profound Thoughts 17

2.3 Differentiability 20

2.4 Weierstrass's Proof 22

2.5 The Aftermath 26

2.6 Final Thoughts 29

Chapter 3 Bézier Curves 32

3.1 Bézier's Curve of Curves 32

3.2 Bézier and Bernstein 38

3.3 Bézier and Casteljau 43

3.4 The Story of Lump 45

3.5 The Story of O 46

Chapter 4 The Rectangular Hyperbola 50

4.1 Old Logarithms 50

4.2 A Thorny Problem 53

4.3 Computation 60

4.4 New Logarithms 62

Chapter 5 The Quadratrix of Hippias 66

5.1 Problems of Antiquity 66

5.2 Some Greek Constructions 68

5.3 The Quadratrix and Trisection 75

5.4 The Quadratrix and Circle Squaring 78

Chapter 6 Two Space-Filling Curves 85

6.1 Je Le Vois, Mais Je Ne Le Crois Pas 85

6.2 Peano's Function 90

6.3 Hilbert's Curve 95

6.4 Peano's Curve 101

Chapter 7 Curves of Constant Width 104

7.1 The Reuleaux Triangle … 105

7.2 …And Its Generalizations 110

7.3 And Their Generalization … 117

7.4 A Circle in All but Name? 123

Chapter 8 The Normal Curve 126

8.1 A Fruitful Question 126

8.2 An Answer but Not a Solution 128

8.3 Approximating the Impossible 131

8.4 Error Curves 138

8.5 The Error Curve 144

8.6 The Normal Distribution 151

Chapter 9 The Catenary 155

9.1 A Matter of Symmetry 155

9.2 Historical Errors 157

9.3 The Curve Identified 162

9.4 Hyperbolic Functions 169

9.5 The Chain Inverted 173

9.6 A Bumpy Road 179

Chapter 10 Elliptic Curves 183

10.1 Elliptic Ambiguity 184

10.2 Problems, Problems, Problems 189

10.3 Common Ground 193

10.4 The Congruent Number Problem 196

10.5 An Arithmetic 203

10.6 Fertile Fields 209

10.7 Cryptography 220

10.8 Apologia 225

Perhaps the Most Important Curve of All 227

Appendix A The Title Page 229

Appendix B Conics Encapsulated 232

Appendix C A Trigonometric Variant for the Bézier Curve 234

Appendix D Envelopes 236

Appendix E The Mathematics of an Arch 240

Appendix F The Simple Pendulum 242

Appendix G Fibonacci's Method 243

References 247

Index 255

What People are Saying About This

From the Publisher

"Havil has done a wonderful job of choosing ten of the most intriguing curves in mathematics—some are famous, some are infamous, and some will likely be new to many readers. I enjoyed this book very much."—David S. Richeson, author of Euler's Gem: The Polyhedron Formula and the Birth of Topology

"Julian Havil enlivens the mathematics of curves with historical context and charming anecdotes, from Galileo's quest for the engineering principle of the arch to Picasso's curvaceous drawing of a dog."—Lynn Gamwell, author of Mathematics and Art: A Cultural History

"A delightful collection of plane curves with engaging historical commentary. The curves range from classical to neoclassical to modern. Some are mainly of theoretical interest, but many have practical applications. All are fascinating."—Avner Ash, coauthor of Elliptic Tales: Curves, Counting, and Number Theory

"With his engaging style of writing, Havil takes ten mathematical curves and describes their properties and the interesting histories behind them. Curves for the Mathematically Curious is an important book."—Allan McRobie, author of The Seduction of Curves: The Lines of Beauty That Connect Mathematics, Art, and the Nude

"With this charming collection of episodes, Havil shows that the study of curves is far from one-dimensional. Ten carefully selected curves provide focus for chapters that abound with mathematical and historical substance as well as practical application. The featured curves serve as a cabinet of wonders for reading straight through or dipping into for a rich treat."—Frank A. Farris, author of Creating Symmetry: The Artful Mathematics of Wallpaper Patterns

Editorial Reviews

"A wonderful addition to libraries where the mathematically curious find their reading."

Choice

"This is not your father’s – or grandfather’s – standard collection of conic sections."—-Jim Stein, New Books Network

"Undoubtedly [this book], written in the same entertaining unmistakable style of the author and containing a lot of information - mathematical, historical and general - will attract, as the previous ones, a large audience."—-S. Cobzas, Studia Mathematica

"What a beautiful book!"—-Jonathan Shock, Mathemafrica.org

From the Publisher