Convex Bodies which Tile Space

  • P. McMullen
Conference paper


We say that the convex body (compact convex set with nonempty interior) K tiles d-dimensional Euclidean space E d (by translation) if there is some family T of translation vectors, such that (i) K covers E d , and (ii) if t i T (i = 1,2) with t 1t 2, then K + t 1 and K + t 2 have disjoint interiors; that is, K is simultaneously a covering and packing of E d . We call K a tiling of E d (by translation), and call K and its translates in K tiles. A particularly important case is when T is a lattice (discrete additive subgroup of E d ), when we call K a lattice tiling.


Convex Body Translation Vector Finite Union Star Body Dimensional Face 
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  1. [1]
    \( Baranovski\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} \), E. P. and Ryškov, S. S., Primitive five-dimensional parallelohedra. Dokl Akad. Nauk SSSR212 (1973), 532–535 = Soviet Math. Dokl.14 (1973), 1391–1395 (1974).MathSciNetGoogle Scholar
  2. [2]
    Coxeter, H. S. M., The classification of zonohedra by means of projective diagrams. J. Math. Pures Appl. 41 (1962), 137–156.MathSciNetMATHGoogle Scholar
  3. [3]Delaunay (Delone), B. N., Sur la partition regulière de l’espace à 4 dimensions, I, II. Izvestia Akad. Nauk SSSR, Ser. VII (1929), 79–110, 147-164.Google Scholar
  4. [4]
    Fedorov, E. S., Elements of the Study of Figures (in Russian). St. Petersburg 1885 (Leningrad 1953).Google Scholar
  5. [5]
    Groemer, H., Ueber Zerlegungen des Euklidischen Raumes. Math. Z. 79 (1962), 364–375.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Groemer, H., Ueber die Zerlegungen des Raumes in homothetische konvexe Körper. Monatsh. Math. 68 (1964), 21–32.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Hilbert, D., Problèmes futurs des mathematiques. In Proc. II Internal Congr. Math. 1900. Paris, 1902.Google Scholar
  8. [8]
    McMullen, P., Space tiling zonotopes. Mathematika 22 (1975), 202–211.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    McMullen, P., Convex bodies which tile space by translation Mathematika 27 (1980), 113–121.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Minkowski, H., Allgemeine Lehrsätze über konvexen Polyeder. Nachr. K. Akad. Wiss. Göttingen, Math.-Phys. Kl. ii (1897), 198–219.Google Scholar
  11. [11]
    Shephard, G. C., Space filling zonotopes. Mathematika 21 (1974), 261–269.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Stein, S. K., A symmetric star body that tiles but not as a lattice. Proc. Amer. Math. Soc. 36 (1972), 543–548.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Voronoi, G. F., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième Mémoire: Recherche sur les paralléloèdres primitifs. J. Reine Angew. Math. 134 (1908), 198–287; 136 (1909), 67–181.MATHCrossRefGoogle Scholar
  14. [14]
    \( \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over Z} itomirski\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} \), O. K., Verschärfung eines Satzes von Woronoi. Ž. Leingr. fiz.-mat. Obšč.2 (1929), 131–151.Google Scholar
  15. [15]
    Aleksandrov, A. D., On filling of space by polytopes (in Russian). Vestnik Leningrad. Univ. (Ser. Mat. Fiz. Him.) 9 (1954), 33–43.MathSciNetGoogle Scholar
  16. [16]
    McMullen, P., Convex bodies which tile space by translation: Acknowledgment of priority. Mathematika 28 (1981).Google Scholar
  17. [17]
    Venkov, B. A., On a class of euclidean polytopes (in Russian). Vestnik Leningrad. Univ. (Ser. Mat. Fiz. Him.) 9 (1954), 11–31.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1981

Authors and Affiliations

  • P. McMullen
    • 1
  1. 1.University CollegeLondonEngland

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