The Symmetries of Things by John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss

This book is one of the rare books that covers the topic of symmetry in a comprehensive and accessible way. It is written by three experts in the field and includes many illustrations and examples to help readers understand the concepts. The book covers a wide range of topics, including symmetry groups, tilings, colorings of patterns, and the classification of patterns to list a few. It is a great resource for anyone interested in the mathematics of symmetry.

While most books approach symmetry primarily through group theory, this one takes a dual perspective, exploring symmetry through both topology and group theory. It serves as an excellent resource for anyone interested in the mathematics of symmetry and its applications in art and design. The book embodies the unique and unmistakable style of the late English mathematician John Horton Conway (December 26, 1937 – April 11, 2020), renowned for his playful and engaging approach to mathematics.

Table of Contents

Part I: Symmetries of Finite Objects and Plane Repeating Patterns

1. Symmetries
2. Planar Patterns
3. The Magic Theorem
4. The Spherical Patterns
5. The Seven Types of Frieze Patterns
6. Why the Magic Theorems Work
7. Euler’s Map Theorem
8. Classification of Surfaces
9. Orbifolds

Part II: Color Symmetry, Group Theory, and Tilings

1. Presenting Presentations
2. Twofold Colorations
3. Threefold Colorings of Plane Patterns
4. Other Primefold Colorings
5. Searching for Relations
6. Types of Tilings
7. Abstract Groups

Part III: Repeating Patterns in Other Spaces

1. Introducing Hyperbolic Groups
2. More on Hyperbolic Groups
3. Archimedean Tilings
4. Generalized Schlafli Symbols
5. Naming Archimedean and Catalan Polyhedra and Tilings
6. The 35 “Prime” Space Groups
7. Objects with Prime Symmetry
8. Flat Universes
9. The 184 Composite Space Groups
10. Higher Still

Appendix

• Other Notations for the Plane and Spherical Groups

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