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Distances on Lozenge Tilings

  • Olivier Bodini
  • Thomas Fernique
  • Éric Rémila
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)

Abstract

In this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the the number of flips – a local transformation on tilings – necessary to link two tilings. We prove that the flip-distance between two tilings is equal to the Hamming-distance for n ≤ 4. We also show, by providing a pair of so-called deficient tilings, that this does not hold for n ≥ 6. We finally discuss the n = 5 case, which remains open.

Keywords

Height Function Local Transformation Bruhat Order Penrose Tiling Tiling Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Arnoux, P., Berthé, V., Fernique, T., Jamet, D.: Functional stepped surfaces, flips and generalized substitutions. Theor. Comput. Sci. 380, 251–265 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented matroids. Encyclopedia of Mathematics, vol. 46. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  3. 3.
    Bodini, O., Latapy, M.: Generalized Tilings with height functions. Morfismos 7, 47–68 (2003)Google Scholar
  4. 4.
    Bodini, O., Fernique, T., Rémila, É.: Characterizations of flip-accessibility for rhombus tilings of the whole plane. Info. & Comput. 206, 1065–1073 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chavanon, F., Rémila, É.: Rhombus tilings: decomposition and space structure. Disc. & Comput. Geom. 35, 329–358 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Bruijn, N.G.: Dualization of multigrids. J. Phys. Fr. C3 47, 9–18 (1986)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Desoutter, V., Destainville, N.: Flip dynamics in three-dimensional random tilings. J. Phys. A: Math. Gen. 38, 17–45 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Felsner, S., Weil, H.: A theorem on higher Bruhat orders. Disc. & Comput. Geom. 23, 121–127 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grnbaum, B., Shephard, G.C.: Tilings and patterns. W. H. Freeman, New York (1986)Google Scholar
  10. 10.
    Kenyon, R.: Tiling a polygon with parallelograms. Algorithmica 9, 382–397 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rémila, É.: On the lattice structure of the set of tilings of a simply connected figure with dominoes. Theor. Comput. Sci. 322, 409–422 (2004)CrossRefzbMATHGoogle Scholar
  12. 12.
    Thurston, W.P.: Conway’s tiling group. American Math. Month. 97, 757–773 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Olivier Bodini
    • 1
  • Thomas Fernique
    • 2
  • Éric Rémila
    • 3
  1. 1.LIP6CNRS & Univ. Paris 6ParisFrance
  2. 2.LIFCNRS & Univ. de ProvenceMarseilleFrance
  3. 3.LIPCNRS & Univ. de Lyon 1LyonFrance

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