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The Symmetries of Things


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Table of Contents

I Symmetries of Finite Objects and Plane Repeating Patterns

1. Symmetries



Rosette Patterns

Frieze Patterns

Repeating Patterns on the Plane and Sphere

Where Are We?

2. Planar Patterns

Mirror Lines

Describing Kaleidoscopes


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?


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?


5. Frieze Patterns

Where Are We?


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?


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


Alias and Alibi

Where Are We?


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


Relative versus Absolute

Enumerating the Tessellations

Archimedes Was Right!

The Hyperbolic Archimedean Tessellations

Examples and Exercises

20. Generalized Schlafli 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


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


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


The Gosset Series

The Symmetries of Still Higher Things

Where Are We?

Other Notations for the Plane and Spherical Groups



About the Author

John H. Conway is the John von Neumann Chair of Mathematics at Princeton University. He obtained his BA and his PhD from the University of Cambridge (England). He is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory, and coding theory. He has also contributed to many branches of recreational mathematics, notably the invention of the Game of Life. Heidi Burgiel is a professor in the Department of Mathematics and Computer Science at Bridgewater State College. She obtained her BS in Mathematics from MIT and her PhD in Mathematics from the University of Washington. Her primary interests are educational technology and discrete geometry. Chaim Goodman-Strauss is a professor in the department of mathematical sciences at the University of Arkansas. He obtained both his BS and PhD in Mathematics at the University of Texas at Austin. His research interests include low-dimensional topology, discrete geometry, differential geometry, the theory of computation, and mathematical illustration. Since 2004 he has been broadcasting mathematics on a weekly radio segment.


The book contains many new results. ... [and] is printed on glossy pages with a large number of beautiful full-colour illustrations, which can be enjoyed even by non-mathematicians. -- EMS Newsletter, June 2009 One of the most base concepts of art [is] symmetry. The Symmetries of Things is a guide to this most basic concept showing that even the most basic of things can be beautiful-and addresses why the simplest of patterns mesmerizes humankind and the psychological and mathematical importance of symmetry in ones every day life. The Symmetries of Things is an intriguing book from first page to last, highly recommended to the many collections that should welcome it. -- The Midwest Book Review, June 2008 Conway, Burgiel, and Goodman-Strauss have written a wonderful book which can be appreciated on many levels. ... [M]athematicians and math-enthusiasts at a wide variety of levels will be able to learn some new mathematics. Even better, the exposition is lively and engaging, and the authors find interesting ways of telling you the things you already know in addition to the things you don't. -- Darren Glass, MAA Reviews, July 2008 This rich study of symmetrical things ... prepares the mind for abstract group theory. It gets somewhere, it justifies the time invested with striking results, and it develops ... phenomena that demand abstraction to yield their fuller meaning... the fullest available exposition with many new results. -- D. V. Feldman, CHOICE Magazine , January 2009 This book is a plaything, an inexhaustible exercise in brain expansion for the reader, a work of art and a bold statement of what the culture of math can be like, all rolled into one. Like any masterpiece, The Symmetries of Things functions on a number of levels simultaneously... It is imperative to get this book into the hands of as many young mathematicians as possible. And then to get it into everyone else's hands. -- Jaron Lanier, American Scientist, January 2009 You accompany the authors as they learn about the structures they so beautifully illustrate on over 400 hundred glossy and full-colour pages. Tacitly, you are given an education in the ways of thought and skills of way-finding in mathematics... The style of writing is relaxed and playful ... we see the fusing of the best aspects of textbooks--conciseness, flow, reader-independence--with the best bit of popular writing--accessibility, fun, beauty. -- Phil Wilson, Plus Magazine, February 2009 This book gives a refreshing and comprehensive account of the subject of symmetry--a subject that has fascinated humankind for centuries... Overall, the book is a treasure trove, full of delights both old and new. Much of it should be accessible for anyone with an undergraduate-level background in mathematics, and is likely to stimulate further interest. -- Marston Conder, Mathematical Reviews, March 2009 Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, together with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments. This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers. -- L'Enseignement Mathematique, December 2009

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