Periodic tilings of regular polygons have been present in history for a very long time: squares and triangles tessellate the plane in a known simple way, tiles and mosaics surround us, hexagons appear in honeycombs and graphene structures. The oldest registry of a systematic study of tilings of the plane with regular polygons is Kepler’s book Harmonices Mundi, published 400 years ago.
In this thesis, we describe a simple integer-based representation for periodic tilings of regular polygons using complex numbers. This representation allowed us to acquire geometrical models from two large collections of images – which constituted the state of the art in the subject –, to synthesize new images of the tilings at any scale with arbitrary precision, and to recognize symmetries and classify each tiling in its wallpaper group as well as in its n-uniform k-Archimedean class.
In this work, we solve the age old problem of characterizing all triangle and square tilings (Sommerville, 1905), and we set the foundations for the enumeration of all periodic tilings with regular polygons. An algebraic structure for families of triangle-square tilings arises from their representation via equivalence with edge-labeled hexagonal graphs. The set of tilings whose edge-labeled hexagonal dual graph is embedded in the same flat torus is closed by positive- integer linear combinations. We compute Hilbert basis for families of tilings in each topological setting. The bases provide the enumeration of the infinite families of tilings spanned by them. Since tilings of triangles and squares contain all other tilings by regular polygons (with exactly one exception), we set the grounds for the enumeration of all periodic tilings with regular polygons. We use generators in the bases to create a sample set of more than 100 million triangle-square tilings, and we describe their general properties and some asymptotic behaviors. Additionally, we show an interpretation of the algebraic structure of triangle-square tilings as origami foldings.