The Symmetries of Things

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.

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