THE ART OF MATHEMATICS
1 Introduction to Symmetry 1 1A Counting Symmetries 1 1B Objects 4 1C Rigid Motions 9 1D Bounded and Unbounded Objects (Revisited) 14 1E Elements of Mathematics: The Contrapositive 17 Exercises 19 2 The Algebra of Symmetry 29 2A Composition 29 2B Cayley Tables 31 2C Groups 34 2D Symmetry Groups 37 2E One Reflection Is Enough 39 2F An Improved Classification of Rigid Motions 41 2G Elements of Mathematics: The Counterexample 45 Exercises 47 3 The Classification Theorems 55 3A Rigid Equivalence 56 3B Bounded Objects 57 3C Border Patterns 60 3D Wallpaper Patterns (optional) 63 3E Elements of Mathematics: Equivalence (optional) 67 Exercises 70 4 Isomorphic Groups 79 4A The Definition of an Isomorphism 80 4B Isomorphism Examples 84 4C A Better Notation for Cyclic Groups 87 4D Elements of Mathematics: The Converse 89 Exercises 90 5 Subgroups and Product Groups 95 5A Subgroups 95 5B Generated Subgroups (optional) 99 5C Product Groups (optional) 101 5D Elements of Mathematics: Sets versus Lists 106 Exercises 107 6 Permutation Groups 111 6A Counting Permutations 111 6B Cycle Notation and Composition 114 6C Even and Odd Permutations 117 6D Symmetries and Permutations 122 6E Elements of Mathematics: Well-Defined 124 6F Elements of Mathematics: The Inductive Proof (optional) 127 Exercises 129 7 Symmetries of 3D Objects 135 7A Basics of 3D 135 7B Essentially Two-Dimensional Objects 140 7C A Trick for Counting Symmetries 143 7D The Tetrahedron 145 7E The Cube 147 7F The Dodecahedron 149 7G The Classification Theorem 150 7H Chirality 153 7I The Full Story (optional) 156 Exercises 163 8 The Five Platonic Solids 171 8A The Classification of Platonic Solids 171 8B Counting Their Parts 176 8C Duality 177 8D Euler’s Formula 180 8E The Euler Characteristic 184 8F The Platonic Solids Through the Ages 185 Exercises 187 9 Symmetry and Optimization 191 9A Minimal Surfaces 191 9B The Circle Wins 193 9C Elements of Mathematics: Proof by Contradiction 197 Exercises 198 10 What Is a Number? 201 10A Natural Numbers 202 10B Integers and Rational Numbers 203 10C Irrational Numbers 203 10D Real Numbers 205 10E Which Real Numbers Are Rational? 207 10F Real Numbers and Symmetry 209 10G Elements of Mathematics: Construction of Rational and Real Numbers 211 Exercises 213 11 Excursions in Numbers 217 11A How Many Prime Numbers Are There? 217 11B The Meaning of “Same Size” 220 11C Are the Rational Numbers Countable? 224 11D Cantor’s Theorem 225 Exercises 228 12 Rigid Motions as Functions 231 12A Measuring Distance in Euclidean Space 231 12B Naming the Points on the Unit Circle 234 12C The Dot Product and Perpendicularity 236 12D Using the Dot Product to Find a Friend 238 12E Rigid Motions Are Functions 239 Exercises 241 13 Rigid Motions as Matrices 243 13A Matrix Computations 243 13B Representing Rigid Motions as Matrices 246 13C Orthogonal Matrices 249 13D Concluding Remarks 250 Exercises 251