A galaxy of periodic tilings

This talk has four parts. In the first part - topology -, we recall the classification of two dimensions manifolds and orbifolds, and introduce their Conway notation. In the second part - geometry -, we discuss two-dimensional periodic tilings and their classification. In the third part - combinatorics -, we introduce Delaney-Dress symbols and indicate how they can be used to systematically enumerate periodic tilings. Finally, in the fourth part - algorithms and software -, we present Tegula, a computer program that can be used to explore the “galaxy” of all 2.4 billion periodic tilings of the euclidean plane, sphere and hyperbolic plane, of low Dress complexity. This is joint work with Olaf Delgado and Rüdiger Zeller.