Plane square tilings

TL;DR

This paper explores the mathematical structure of plane square tilings, establishing a correspondence with harmonic vectors on plane maps, and characterizes periodic tilings using topological and symmetry considerations.

Contribution

It extends a formalism linking square tilings to harmonic vectors and describes the classification of periodic tilings via homology and symmetry analysis.

Findings

Periodic tilings correspond to harmonic vectors on a torus.

The space of harmonic vectors is isomorphic to the first homology group of a torus.

A method for enumerating regular periodic tilings with a given number of square orbits is proposed.

Abstract

We consider here square tilings of the plane. By extending the formalism introduced in [3] we build a correspondence between plane maps endowed with an harmonic vector and square tilings satisfying a condition of regularity. In the case of periodic plane square tiling the relevant space of harmonic vectors is actually isomorphic to the first homology group of a torus. So, periodic plane square tilings are described by two parameters and the set of parameters is split into angular sectors. The correspondence between symmetry of the square tiling and symmetry of the plane maps and harmonic vectors is discussed and a method for enumerating the regular periodic plane square tilings having r orbits of squares is outlined.