Constructing and Visualizing Uniform Tilings

Abstract

This paper describes a system which takes user input of a pattern of regular polygons around one vertex and attempts to construct a uniform tiling with the same pattern at every vertex by adding one polygon at a time. The system constructs spherical, planar, or hyperbolic tilings when the sum of the interior angles of the user-specified regular polygons is respectively less than, equal to, or greater than 360∘. Other works have catalogued uniform tilings in tables and/or illustrations. In contrast, this system was developed as an interactive educational tool for people to learn about symmetry and tilings by trial and error through proposing potential vertex patterns and investigating whether they work. Users can watch the rest of the polygons being automatically added one by one with recursive backtracking. When a trial polygon addition is found to violate the conditions of a regular tiling, polygons are removed one by one until a configuration with another compatible choice is found, and that choice is tried next.