The Symmetries of Things

Produktbeschreibung

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. 

Inhaltsverzeichnis

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 

Autoreninfo

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.