Translational tilings by a polytope, with multiplicity

Abstract

We study the problem of covering ℤd by overlapping translates of a convex polytope, such that almost every point of ℤd is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov [5] and Minkowski [15], and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile ℤd.

By contrast, for k ≥2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles ℤd by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational polytope, is centrally symmetric, and has centrally symmetric facets, then P must k-tile ℤd for some positive integer k.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    A. D. Alexandrov: Convex polyhedra, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad (1950), Akademie-Verlag, Berlin (1958), reprinted in English, by Springer-Verlag, Berlin (2005).

    Google Scholar 

  2. [2]

    R. Diaz, Quang-Nhat Le, and S. Robins: A Fourier-analytic approach to solid angle sums over polytopes, preprint.

  3. [3]

    B. Gordon: Mutiple tilings of Euclidean space by unit cubes, Computers and Mathematics with Applications 39 (2000), 49–53.

    MATH  Article  Google Scholar 

  4. [4]

    H. M. S. Coxeter: Regular polytopes, 2nd ed., MacMillan, New York (1963), 3rd ed., Dover, New York (1973).

    Google Scholar 

  5. [5]

    E. S. Fedorov: The symmetry of regular systems of figures, Zap. Miner. Obshch. 21 (1885), 1–279; in Russian.

    Google Scholar 

  6. [6]

    P. Furtwängler: Über Gitter konstanter Dichte, Monatsh. Math. Phys. 43, (1936), 281–288.

    MathSciNet  Article  Google Scholar 

  7. [7]

    P. M. Gruber: Convex and discrete geometry, Springer, Berlin, (2007), 1–589.

    Google Scholar 

  8. [8]

    G. Hajós: Über einfache und mehrfache Bedeckung des n-dimensionalen Raumes mit einem Würfelgitter, Math. Z. 47, (1942), 427–467.

    Article  Google Scholar 

  9. [9]

    M. N. Kolountzakis: On the structure of multiple translational tilings by polygonal regions, Discrete Comput. Geom. 23 (2000), 537–553.

    MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    M. N. Kolountzakis: The study of translational tiling with Fourier Analysis, Lectures given at the Workshop on Fourier Analysis and Convexity, Universita di Milano-Bicocca, June 11–22, (2001).

  11. [11]

    M. N. Kolountzakis and M. Matolcsi: Tilings by translation, La Gaceta de la Real Sociedad Espanola 13 (2010).

  12. [12]

    J. C. Lagarias and Y. Wang: Tiling the line with translates of one tile, Inventiones math. 124 (1996), 341–365.

    MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    P. McMullen: Convex bodies which tile space by translation, Mathematika 27 (1980), 113–121.

    MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    P. McMullen: Space tiling zonotopes, Mathematika 22 (1975), 202–211.

    MathSciNet  MATH  Article  Google Scholar 

  15. [15]

    H. Minkowski: Allgemeine Lehrsätze über die convexen Polyeder, Nachr. Ges. Wiss. Göttingen 1897, 198–219.

  16. [16]

    R. Robinson: Multiple tilings of n-dimensional space by unit cubes, Mathematische Zeischrift 166 (1979), 225–264.

    MATH  Article  Google Scholar 

  17. [17]

    E. M. Stein and G. Weiss: Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series 32 (1971), 1–297.

    MathSciNet  Google Scholar 

  18. [18]

    B. A. Venkov: On a class of Euclidean polyhedra, Vestnik Leningrad Univ. Ser. Math. Fiz. Him. 9 (1954), 11–31; in Russian.

    MathSciNet  Google Scholar 

  19. [19]

    U. Bolle: On multiple tiles in2, Intuitive Geometry (1994).

  20. [20]

    G. Ziegler: Lectures on Polytopes, Springer Graduate Texts in Mathematics (1995), 1–365.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sinai Robins.

Additional information

The second author is grateful for the support of the Ministry of Education, Singapore, research grant MOE2011-T2-1-090

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gravin, N., Robins, S. & Shiryaev, D. Translational tilings by a polytope, with multiplicity. Combinatorica 32, 629–649 (2012). https://doi.org/10.1007/s00493-012-2860-3

Download citation

Mathematics Subject Classification (2000)

  • 52B11
  • 52C07
  • 52C17
  • 52C22