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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.

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References

  1. Bölcskei, A.: Classification of unilateral and equitransitive tilings by squares of three sizes. Beiträge Algebra Geom. 41, 267–277 (2000)

    MATH  Google Scholar 

  2. Bölcskei, A.: Filling space with cubes of two sizes. Publ. Math. Debrecen 59, 317–326 (2001)

    MathSciNet  MATH  Google Scholar 

  3. 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)

    MathSciNet  Google Scholar 

  4. 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)

    Article  MathSciNet  Google Scholar 

  5. Dehn, M.: Zerlegung von Rechtecke in Rechtecken. Math. Annalen 57, 314–332 (1903)

    Article  MathSciNet  MATH  Google Scholar 

  6. Delgado, O., Huson, D., Zamorzaeva, E.: The classification of \(2\)-isohedral tilings of the plane. Geometriae Dedicata 42, 43–117 (1992)

    Google Scholar 

  7. 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)

    Google Scholar 

  8. Gambini, I.: Quant aux carrés carrelés. PhD thesis Université de la Méditerranée (1999)

  9. Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W. H. Freeman, New York (1987)

    MATH  Google Scholar 

  10. Kenyon, R.: Tilings and discrete Dirichlet problems. Israel J. Math. 105, 61–84 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Martini, H., Makai, E., Soltan, V.: Unilateral tilings of the plane with squares of three sizes. Beiträge Algebra Geom. 39, 481–495 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Nakahara, M.: Geometry, Topology and Physics. Institute of physics, Bristol (2003)

    Book  MATH  Google Scholar 

  13. Schattschneider, D.: Unilateral and equitransitive tilings by squares. Discrete Comput. Geom. 24, 519–525 (2000)

    Google Scholar 

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Acknowledgments

We thank the referee for indicating us the notion of discrete harmonic function and that the first version of Lemma 1 was incorrect.

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Authors and Affiliations

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

    Mathieu Dutour Sikirić

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  1. Mathieu Dutour Sikirić
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Correspondence to Mathieu Dutour Sikirić.

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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.

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Dutour Sikirić, M. Torus square tilings. AAECC 23, 251–261 (2012). https://doi.org/10.1007/s00200-012-0178-4

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  • DOI: https://doi.org/10.1007/s00200-012-0178-4

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