Regular production systems and triangle tilings

Abstract

We discuss regular production systems as a tool for analyzing tilings in general. As an application we give necessary and sufficient conditions for a generic triangle to admit a tiling of ${\mathbb{H}}^2$ and show that almost every triangle that admits a tiling is “weakly aperiodic”. We pause for informal discussion of a variety of other applications, such as non-quasi-isometric maps between regular tilings, non-periodic Archimedean tilings, growth, and decidability. Most generally, regular production systems provide a model for the organized growth of surfaces along a front, subject to local rules.