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Planar Configurations Induced by Exact Polyominoes

  • Daniela Battaglino
  • Andrea Frosini
  • Simone Rinaldi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6636)

Abstract

An unknown planar discrete set of points A can be inspected by means of a probe P of generic shape that moves around it, and reveals, for each position, the number of its elements as a magnifying glass. All the data collected during this process can be naturally arranged in an integer matrix that we call the scan of the starting set A w.r.t. the probe P.

When the probe is a rectangle, a set A whose scan is homogeneous shows a strong periodical behavior, and can be decomposed into smaller homogeneous subsets. Here we extend this result, which has been conjectured true for all the exact polyominoes, to the class of diamonds, and we furnish experimental evidence of the decomposition theorem for exact polyominoes of small dimension, using the mathematical software Sage.

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References

  1. 1.
    Beauquier, D., Nivat, M.: On Translating one polyomino to tile the plane. Discrete Comput. Geom. 6, 575–592 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brleck, S., Frosini, A., Rinaldi, S., Vuillon, L.: Tilings by translation: enumeration by a rational language approach. The Electronic Journal of Combinatorics 13(1) (2006)Google Scholar
  3. 3.
    Brlek, S., Provençal, X.: An optimal algorithm for detecting pseudo-squares. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 403–412. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Nivat, M.: On a tomographic equivalence between (0,1)-matrices. In: Karhumäki, J., Maurer, H., Păun, G., Rozenberg, G. (eds.) Theory Is Forever. LNCS, vol. 3113, pp. 216–234. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Frosini, A., Nivat, M.: Binary matrices under the microscope: a tomographical problem. TCS 370, 201–217 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Herman, G.T., Kuba, A.: Discrete tomography: Foundations algorithms and applications. Birkhauser, Boston (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    Nivat, M.: Sous-ensembles homogenes de \(\mathbb{z}^2\) et pavages du plan. C. R. Acad. Sci. Paris. I 335, 83–86 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Provençal, X.: Combinatoire des mots, geometrie discrete et pavages. Ph.D. thesis, D1715, Université du Québec a Montréal (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Daniela Battaglino
    • 1
  • Andrea Frosini
    • 2
  • Simone Rinaldi
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversità di SienaSienaItaly
  2. 2.Dipartimento di Sistemi e InformaticaUniversità di FirenzeFirenzeItaly

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