Torus square tilings
 Mathieu Dutour Sikirić
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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.
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 Title
 Torus square tilings
 Journal

Applicable Algebra in Engineering, Communication and Computing
Volume 23, Issue 56 , pp 251261
 Cover Date
 20121201
 DOI
 10.1007/s0020001201784
 Print ISSN
 09381279
 Online ISSN
 14320622
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Square tiling
 Harmonic vectors
 Hodge theory
 Electrical network
 05B45
 52C20
 Industry Sectors
 Authors

 Mathieu Dutour Sikirić ^{(1)}
 Author Affiliations

 1. Rudjer Bosković Institute, Bijenicka 54, 10000, Zagreb, Croatia