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4 With love to Anu who understands it all ... I am indebted to the many mathematicians of the past and present from whose ideas this book has been distilled. I am grateful to Marty Ross and John Stillwell for their criticism and insightful comments. Finally, many many thanks to John Martineau and Daud Sutton for being patient guides and accomplices in opening this visual vortex into the beautiful world of mathematical proofs. Two infinite sums boxed and ready to be served. CONTENTS Introduction 1 Treacherous Truth 2 Pythagoras Theorem 4 Plane and Simple 6 From Pie to Pi 8 Cavalieris Principle 10 Cavalier Cone Carving 12 A Frustrating Frustum 14 Archimedes Theorem 16 Inside Out 18 Mathematical Dominos 20 The Infinite Staircase 22 Circling the Cycloid 24 Slicing Cones 26 Folding Conics 28 Knotting Polygons 30 Cutting Squares 32 Power Sums 34 Neverending Primes 36 The Nature of Numbers 38 The Golden Ratio 40 The Numbers of Nature 42 Eulers Formula 44 Possible Impossibilities 46 Appendix I One Theorem, Many Proofs 48 Appendix II All for One and One for All 50 Appendix III Looks can be Deceiving 52 Appendix IV Triangles of Generality 54 Appendix V Polytopes of Analogy 56
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