Periodicity of one-dimensional tilings

  • V. Sidorenko
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 829)


The tiling problem is closely connected with a number of problems of information theory. We show that the Wang-Moore conjecture is valid for the one dimensional case. Namely, any set of templates which permits a tiling of the set ℤ of integers also permits a periodic tiling. We also show that any tiling of ℤ by one template is necessarily periodic. The obtained results are also valid for tiling of the set IR of real numbers.


  1. 1.
    V. Sidorenko, Tilings of the plane and codes for translational combinatorial metrics, accepted to IEEE Int. Symp. on Inf. Theory, Trondheim, 1994Google Scholar
  2. 2.
    H. D. Shapiro, Theoretical limitations on the efficient use of parallel memories. IEEE Trans. Comput., vol. C-27, 421–428, May 1978Google Scholar
  3. 3.
    E. Belitskaya, V. Sidorenko and P. Stenström, Testing of memory with defects of fixed configuration. II Int. workshop Algebraic and combinatorial coding theory, Proc., Leningrad, USSR, 24–28, Sept. 1990Google Scholar
  4. 4.
    S. W. Golomb, Polyominos. Allen and Unwin, London, 1966Google Scholar
  5. 5.
    H. Wang, Proving theorems by pattern recognition — II. Bell System Tech. J., 40, 1–41, 1961Google Scholar
  6. 6.
    R.Berger, The undecidability of the domino problem. Mem. Amer. Math. Soc., 66, 1966Google Scholar
  7. 7.
    R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane. Invent. Math., 12, 177–209, 1971Google Scholar
  8. 8.
    H.A.G. Wijshoff and J.van Leuwen, Arbitrary versus periodic storage schemes and tessellations of the plane using one type of polyomino. Inform. and Control, vol. 62, 1–25, May 1984Google Scholar
  9. 9.
    D. Beauquier and M. Nivat, On translating one polyomino to tile the plane. Discrete Comput. Geom., 6, 575–592, 1991Google Scholar
  10. 10.
    C. Radin, Tiling, periodicity, and crystals. J. Math. Phys., 26(6), 1342–1344, June 1985Google Scholar
  11. 11.
    C. Radin and L. S. Schulman, Periodicity of classical ground states. Phys. Review Letters, 51(8), 621–622, Aug. 1983Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • V. Sidorenko
    • 1
  1. 1.Institute for Problems of Information TransmissionMoscowRussia

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