Torus square tilings

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Abstract

We consider periodic square tilings of the plane. By extending a formalism introduced in 1940 for tiling of rectangles by squares we build a correspondence between periodic plane maps endowed with a periodic harmonic vector and periodic square tilings satisfying a regularity condition. The space of harmonic vectors is 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 corresponding plane map and harmonic vector is discussed and a method for enumerating the regular periodic plane square tilings having $r$ orbits of squares is outlined.

The author has been supported by the Croatian Ministry of Science, Education and Sport under contract 098-0982705-2707. The author also thanks Y. Itoh for having invited him in Hayama where this research was initiated.