Symmetry: A Mathematical Exploration

GetFullPageImage......Page 1
front-matter......Page 2
Symmetry......Page 4
Preface......Page 6
Table of Notation......Page 10
Table of Contents......Page 12
1. Introduction to Symmetry......Page 16
A Precise Definition of “Symmetry”......Page 17
Types of Symmetries and Types of Objects......Page 21
The Classification of Plane Rigid Motions......Page 27
(1) Which are proper and which are improper:......Page 29
2. The Algebra of Symmetry......Page 32
Cayley Tables......Page 33
Symmetry Groups......Page 36
The Power of Inverses......Page 40
3. Isomorphism......Page 49
Rigid Equivalence......Page 57
A Better Notation for the Cyclic Groups......Page 60
Exercises......Page 62
Bounded Objects......Page 65
Border Patterns......Page 68
Wallpaper Patterns......Page 70
Summary......Page 73
Exercises......Page 74
5. Subgroups and Product Groups......Page 77
Generating Subgroups......Page 80
Product Groups......Page 83
Exercises......Page 87
6. Permutations......Page 89
Permutation Groups......Page 90
Even and Odd Permutations......Page 94
Exercises......Page 98
7. Symmetries of Solid Objects......Page 101
The Symmetry Group a Tetrahedron......Page 107
The Proper Symmetry Group a Cube......Page 110
The Proper Symmetry Group a Dodecahedron......Page 111
Solid Objects Which Are “Essentially Two-Dimensional”......Page 113
The Classification Theorem for Bounded Objects......Page 115
Chirality......Page 118
Proper Versus Full Symmetry Groups......Page 121
Exercises......Page 125
8. The Five Platonic Solids......Page 129
Counting Their Parts......Page 134
Duality......Page 135
Euler’s Formula......Page 138
The Euler Characteristic......Page 143
An Algebraic Proof that There Are Only Five Platonic 
Solids......Page 145
The Platonic Solids Through the Ages......Page 148
Exercises......Page 149
Minimal Surfaces......Page 152
The Circle Wins......Page 155
Exercises......Page 160
Natural Numbers......Page 162
Rational Numbers......Page 164
Real Numbers......Page 166
Which Real Numbers Are Rational?......Page 169
How Many Primes Are There?......Page 171
Exercises......Page 175
11. Cantor’s Infinity......Page 179
The Modern Meaning of “Same Size”......Page 180
Are the Rational Numbers Countable?......Page 184
Cantor’s Theorem......Page 186
Exercises......Page 189
12. Euclidean Space......Page 191
The Pythagoran Theorem and Distance Formula......Page 192
Naming the Points on the Unit Circle......Page 195
The Dot Product and Perpendicularity......Page 197
Using the Dot Product to Find a Lover or a Song......Page 199
What is a Rigid Motion?......Page 202
Two Exotic Examples......Page 205
Exercises......Page 207
Matrix Computations......Page 210
Representing Rigid Motions as Matrices......Page 213
Orthogonal Matrices......Page 217
You Finished the Book. Now What?......Page 219
Exercises......Page 220
Index......Page 223