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.

Symmetries of Finite Objects and Plane Repeating Patterns

Symmetries

Planar Patterns

The Magic Theorem

The Spherical Patterns

Frieze Patterns

Why the Magic Theorems Work

Euler's Map Theorem

Classification of Surfaces

Orbifolds

Color Symmetry, Group Theory, and Tilings

Presenting Presentations

Twofold Colorations

Threefold Colorings of Plane Patterns

Searching for Relations

Types of Tilings

Abstract Groups

Repeating Patterns in Other Spaces

Introducing Hyperbolic Groups

More on Hyperbolic Groups

Archimedean Tilings

Generalized Schläfli Symbols

Naming Archimedean and Catalan Polyhedra and Tilings

The 35 "Prime" Space Groups

Objects with Prime Symmetry

Flat Universes

The 184 Composite Space Groups

Higher Still