To Infinity and Beyond

Eli Maor is the author of Beautiful Geometry (with Eugen Jost), Venus in Transit, Trigonometric Delights, e: The Story of a Number, and The Pythagorean Theorem: A 4,000-Year History (all Princeton).

Foreword vii

Preface i

Color Insert 43

Part I Mathematical Infinity

1 First Steps to Infinity Zero, One, Infinity 6

2 Towards Legitimation 10

Numbers Large and Small 14

3 Convergence and Limit 17

The Prime Numbers 21

4 The Fascination of Infinite Series 25

5 The Geometric Series 29

6 More about Infinite Series 34

7 Interlude: An Excursion into the Number Concept 40

8 The Discovery of Irrational Numbers 44

A Do-It-Yourself Method for Finding //2 49

Pi, Phi, and e--Three Celebrated Irrationals 50

9 Cantor's New Look at the Infinite 54

10 Beyond Infinity 61

Part II Geometric Infinity

11 Some Functions and Their Graphs 68

Some Geometric Paradoxes Involving Infinity 83

12 Inversion in Circle 88

13 Geographic Maps and Infinity 95

14 Tiling the Plane 102

15 New Look at Geometry 108

16 The Vain Search for Absolute Truth 118

Part III Aesthetic Infinity

17 Rejoice the Infinite! 136

18 The Mobius Scrip 139

19 The Magic World of Mirrors 149

20 Horror Vacui, Amor Infiniti 155

21 Maurits C Escher-Master of the Infinite 164

22 The Modern Kabbalists 19

Part IV Cosmological Infinity

23 The Ancient World 184

24 The New Cosmology 190

25 The Horizons Are Receding 199

26 Paradox and Its Aftermath 204

27 The Expanding Universe 212

28 The Modern Atomises 224

29 Which Way from Here? 227

Epilogue 232

Appendix 235

Bibliography 260

Index 269