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The Symmetries of Things
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Book Description
Start with a single shape. Repeat it in some way—translation, reflection over a line, rotation around a point—and you have created symmetry.
Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, 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.
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
Author(s)
Biography
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.
Reviews
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 2009This 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