A galaxy of periodic tilings.

23^rd January 2024, 14:00
Daniel Huson
University of Tübingen, Germany

Abstract

This talk has four parts. In the first part - topology -, we recall the classification of two-dimensional manifolds and orbifolds, and introduce their Conway-Macbeath 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 (http://tegula.husonlab.org), a computer program that can be used the 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 (https://doi.org/10.1016/j.cagd.2021.102027).