2010/12/31 by Mathieu Dutour Sikirić, Sikirić, Mathieu Dutour
Computer Science · Materials Science · Mathematics · #Cellular Automata and Applications #Combinatorics #Combinatorics (math.CO) #FOS: Mathematics #Geometry #Mathematical analysis #Mathematics #Mathematics and Applications #Plane (geometry) #Plane symmetry #Quasicrystal Structures and Properties #Square (algebra) #Square tiling #Substitution tiling #Torus #math.CO
paper · pdf · doi:10.48550/arxiv.1101.0223
published in arXiv (Cornell University) (Cornell University) · 14 pages, 7 figures
arxiv created 2010/12/31 · openalex publication_date 2010/12/31 · arxiv updated 2011/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.