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# The Symmetries of Things

## byJohn H. Conway, Heidi Burgiel, Chaim Goodman-Strauss

### Hardcover | April 18, 2008

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The following ISBNs are associated with this title:

ISBN - 10:1568812205

ISBN - 13:9781568812205

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Table of Contents

I Symmetries of Finite Objects and Plane Repeating Patterns

1. Symmetries

Kaleidoscopes

Gyrations

Rosette Patterns

Frieze Patterns

Repeating Patterns on the Plane and Sphere

Where Are We?

2. Planar Patterns

Mirror Lines

Describing Kaleidoscopes

Gyrations

More Mirrors and Miracles

Wanderings and Wonder-Rings

The Four Fundamental Features!

Where Are We?

3. The Magic Theorem

Everything Has Its Cost!

Finding the Signature of a Pattern

Just Symmetry Types

How the Signature Determines the Symmetry Type

Interlude: About Kaleidoscopes

Where Are We?

Exercises

4. The Spherical Patterns

The 14 Varieties of Spherical Pattern

The Existence Problem: Proving the Proviso

Group Theory and All the Spherical Symmetry Types

All the Spherical Types

Where Are We?

Examples

5. Frieze Patterns

Where Are We?

Exercises

6. Why the Magic Theorems Work

Folding Up Our Surface

Maps on the Sphere: Euler's Theorem

Why char = ch

The Magic Theorem for Frieze Patterns

The Magic Theorem for Plane Patterns

Where Are We?

7. Euler's Map Theorem

Proof of Euler's Theorem

The Euler Characteristic of a Surface

The Euler Characteristics of Familiar Surfaces

Where Are We?

8. Classification of Surfaces

Caps, Crosscaps, Handles, and Cross-Handles

We Don't Need Cross-Handles

Two crosscaps make one handle

That's All, Folks!

Where Are We?

Examples

9. Orbifolds

II Color Symmetry, Group Theory, and Tilings

10. Presenting Presentations

Generators Corresponding to Features

The Geometry of the Generators

Where Are We?

11. Twofold Colorations

Describing Twofold Symmetries

Classifying Twofold Plane Colorings

Complete List of Twofold Color Types

Duality Groups

Where Are We?

13. Threefold Colorings of Plane Patterns

A Look at Threefold Colorings

Complete List for Plane Patterns

Where Are We?

Other Primefold Colorings

Plane Patterns

The Remaining Primefold Types for Plane Patterns

The "Gaussian" Cases

The "Eisensteinian" Cases

Spherical Patterns and Frieze Patterns

Where Are We?

14. Searching for Relations

On Left and Right

Justifying the Presentations

The Sufficiency of the Relations

The General Case

Simplifications

Alias and Alibi

Where Are We?

Exercises

Answers to Exercises

15. Types of Tilings

Heesch Types

Isohedral Types

Where Are We?

16. Abstract Groups

Cyclic Groups, Direct Products, and Abelian Groups

Split and Non-split Extensions

Dihedral, Quaternionic, and QuasiDihedral Groups

Extraspecial and Special Groups

Groups of the Simplest Orders

The Group Number Function gnu(n)

The gnu-Hunting Conjecture: Hunting moas

Appendix: The Number of Groups to Order 2009

III Repeating Patterns in Other Spaces

17. Introducing Hyperbolic Groups

No Projection Is Perfect!

Analyzing Hyperbolic Patterns

What Do Negative Characteristics Mean?

Types of Coloring, Tiling, and Group Presentations

Where Are We?

18. More on Hyperbolic Groups

Which Signatures Are Really the Same?

Inequivalence and Equivalence Theorems

Existence and Construction

Enumerating Hyperbolic Groups

Thurston's Geometrization Program

Appendix: Proof of the Inequivalence Theorem

Interlude: Two Drums That Sound the Same

19. Archimedean Tilings

The Permutation Symbol

Existence

Relative versus Absolute

Enumerating the Tessellations

Archimedes Was Right!

The Hyperbolic Archimedean Tessellations

Examples and Exercises

20. Generalized Schläfli Symbols

Flags and Flagstones

More Precise Definitions

More General Definitions

Interlude: Polygons and Polytopes

21. Naming Archimedean and Catalan Polyhedra and Tilings

Truncation and "Kis"ing

Marriage and Children

Coxeter's Semi-Snub Operation

Euclidean Plane Tessellations

Additional Data

Architectonic and Catoptric Tessellations

22. The 35 "Prime" Space Groups

The Three Lattices

Displaying the Groups

Translation Lattices and Point Groups

Catalogue of Plenary Groups

The Quarter Groups

Catalogue of Quarter Groups

Why This List Is Complete

Appendix: Generators and Relations

23. Objects with Prime Symmetry

The Three Lattices

Voronoi Tilings of the Lattices

Salt, Diamond, and Bubbles

Infinite Platonic Polyhedra

Their Archimedean Relatives

Pseudo-Platonic Polyhedra

The Three Atomic Nets and Their Septa

Naming Points

Polystix

Checkerstix and the Quarter Groups

Hexastix from Checkerstix

Tristakes, Hexastakes, and Tetrastakes

Understanding the Irish Bubbles

The Triamond Net and Hemistix

Further Remarks about Space Groups

24. Flat Universes

Compact Platycosms

Torocosms

The Klein Bottle as a Universe

The Other Platycosms

Infinite Platycosms

Where Are We?

25. The 184 Composite Space Groups

The Alias Problem

Examples and Exercises

26. Higher Still

Four-Dimensional Point Groups

Regular Polytopes

Four-Dimensional Archimedean Polytopes

Regular Star-Polytopes

Groups Generated by Reflections

Hemicubes

The Gosset Series

The Symmetries of Still Higher Things

Where Are We?

Other Notations for the Plane and Spherical Groups

Bibliography

Index

Editorial Reviews