444 Pages
    by A K Peters/CRC Press

    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.

    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

    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.

    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