Symmetry: A Mathematical Exploration

✍ Scribed by Kristopher Tapp

• Publisher: Springer
• Year: 2011
• Tongue: English
• Leaves: 230
• Edition: 2012
• Category: Library

✦ Synopsis

This textbook is perfect for a math course for non-math majors, with the goal of encouraging effective analytical thinking and exposing students to elegant mathematical ideas.  It includes many topics commonly found in sampler courses, like Platonic solids, Euler’s formula, irrational numbers, countable sets, permutations, and a proof of the Pythagorean Theorem.  All of these topics serve a single compelling goal: understanding the mathematical patterns underlying the symmetry that we observe in the physical world around us. The exposition is engaging, precise and rigorous.  The theorems are visually motivated with intuitive proofs appropriate for the intended audience.  Students from all majors will enjoy the many beautiful topics herein, and will come to better appreciate the powerful cumulative nature of mathematics as these topics are woven together into a single fascinating story about the ways in which objects can be symmetric.

✦ Table of Contents

Cover......Page 1
Symmetry: A Mathematical Exploration......Page 4
Copyright......Page 5
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
Exercises......Page 29
2. The Algebra of Symmetry......Page 32
Cayley Tables......Page 33
Symmetry Groups......Page 36
The Power of Inverses......Page 40
An Improved Classification of Plane Rigid Motions......Page 43
Exercises......Page 45
3. Isomorphism......Page 50
What is an Isomorphism?......Page 51
Isomorphism Examples......Page 54
Rigid Equivalence......Page 58
A Better Notation for the Cyclic Groups......Page 61
Exercises......Page 63
Bounded Objects......Page 66
Border Patterns......Page 69
Wallpaper Patterns......Page 71
Summary......Page 74
Exercises......Page 75
Subgroups......Page 78
Generating Subgroups......Page 81
Product Groups......Page 84
Exercises......Page 88
6. Permutations......Page 90
Permutation Groups......Page 91
Even and Odd Permutations......Page 95
Exercises......Page 99
Rigid Motions of Space......Page 102
The Symmetry Group a Tetrahedron......Page 108
The Proper Symmetry Group a Cube......Page 111
The Proper Symmetry Group a Dodecahedron......Page 112
Solid Objects Which Are “Essentially Two-Dimensional”......Page 114
The Classification Theorem for Bounded Objects......Page 116
Chirality......Page 119
Proper Versus Full Symmetry Groups......Page 122
Exercises......Page 126
8. The Five Platonic Solids......Page 130
Counting Their Parts......Page 135
Duality......Page 136
Euler’s Formula......Page 139
The Euler Characteristic......Page 144
An Algebraic Proof that There Are Only Five Platonic
Solids......Page 146
The Platonic Solids Through the Ages......Page 149
Exercises......Page 150
Minimal Surfaces......Page 154
The Circle Wins......Page 157
Exercises......Page 162
Natural Numbers......Page 164
Rational Numbers......Page 166
Real Numbers......Page 168
Which Real Numbers Are Rational?......Page 171
How Many Primes Are There?......Page 173
Exercises......Page 177
11. Cantor’s Infinity......Page 182
The Modern Meaning of “Same Size”......Page 183
Are the Rational Numbers Countable?......Page 187
Cantor’s Theorem......Page 189
Exercises......Page 192
12. Euclidean Space......Page 194
The Pythagoran Theorem and Distance Formula......Page 195
Naming the Points on the Unit Circle......Page 198
The Dot Product and Perpendicularity......Page 200
Using the Dot Product to Find a Lover or a Song......Page 202
What is a Rigid Motion?......Page 205
Two Exotic Examples......Page 208
Exercises......Page 210
Matrix Computations......Page 214
Representing Rigid Motions as Matrices......Page 217
Orthogonal Matrices......Page 221
You Finished the Book. Now What?......Page 223
Exercises......Page 224
Index......Page 228