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.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Bölcskei, A.: Classification of unilateral and equitransitive tilings by squares of three sizes. Beiträge Algebra Geom. 41, 267–277 (2000)
Bölcskei, A.: Filling space with cubes of two sizes. Publ. Math. Debrecen 59, 317–326 (2001)
Brinkmann, G.: Isomorphism rejection in structure generation programs. Discrete mathematical chemistry, DIMACS Ser. Discret. Math. Theoret. Comput. Sci. Am. Math. Soc. 51, 25–38 (2000)
Brooks, R.L., Smith, C.A.B., Stone, A.H., Tutte, W.T.: The dissection of rectangles into squares. Duke Math. J. 7, 312–340 (1940)
Dehn, M.: Zerlegung von Rechtecke in Rechtecken. Math. Annalen 57, 314–332 (1903)
Delgado, O., Huson, D., Zamorzaeva, E.: The classification of \(2\)-isohedral tilings of the plane. Geometriae Dedicata 42, 43–117 (1992)
Dutour Sikirić, M., Deza, M.: 4-regular and self-dual analogs of fullerenes. Mathematics and Topology of Fullerenes. In: Ori, O., Graovac, A., Catalado, F. (eds.) Carbon Materials, Chemistry and Physics, vol. 4, pp. 103–116. Springer, Berlin (2011)
Gambini, I.: Quant aux carrés carrelés. PhD thesis Université de la Méditerranée (1999)
Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W. H. Freeman, New York (1987)
Kenyon, R.: Tilings and discrete Dirichlet problems. Israel J. Math. 105, 61–84 (1998)
Martini, H., Makai, E., Soltan, V.: Unilateral tilings of the plane with squares of three sizes. Beiträge Algebra Geom. 39, 481–495 (1998)
Nakahara, M.: Geometry, Topology and Physics. Institute of physics, Bristol (2003)
Schattschneider, D.: Unilateral and equitransitive tilings by squares. Discrete Comput. Geom. 24, 519–525 (2000)
Acknowledgments
We thank the referee for indicating us the notion of discrete harmonic function and that the first version of Lemma 1 was incorrect.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
Dutour Sikirić, M. Torus square tilings. AAECC 23, 251–261 (2012). https://doi.org/10.1007/s00200-012-0178-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-012-0178-4