On periodic tilings with regular polygons

Abstract

The thesis describes a simple integer-based computational representation for periodic tilings of regular polygons using complex numbers, which is now the state of the art for these objects. Several properties of this representation are discussed, including elegant and efficient strategies for acquisition, reconstruction, rendering, and automatic crystallographic classification by symmetry detection. The thesis also describes a novel strategy for the enumeration and generation of triangle-square tilings via equivalence with edge-labeled hexagonal graphs. The equivalence provide triangle-square tilings with an algebraic structure that allows an unfolding interpretation.