Skip to main content
    • Aa
    • Aa

Convex bodies which tile space by translation

  • P. McMullen (a1)

It is shown that a convex body K tiles Ed by translation if, and only if, K is a centrally symmetric d-polytope with centrally symmetric facets, such that every belt of K (consisting of those of its facets which contain a translate of a given (d – 2)-face) has four or six facets. One consequence of the proof of this result is that, if K tiles Ed by translation, then K admits a face-to-face, and hence a lattice tiling.

Hide All
Aleksandrov A. D.. “A theorem on convex polyhedra”, Trudy Mat. Inst. Steklov 4 (1933), 87 (Russian).
Aleksandrov A. D.. Konvexe Polyeder (Akademie-Verlag, Berlin, 1958) (translated from Russian edition, 1950).
Coxeter H. S. M.. “The classification of zonohedra by means of projective diagrams”, J. Maths. Pures Appl. 41 (1962), 137156.
Coxeter H. S. M.. Regular polytopes (Dover, 1973).
Delaunay B. N. (Delone). “Sur la théorie des paralléloèdres”, C. R. Acad. Sci. Paris 183 (1926), 464467.
Delaunay B. N.. “Sur la partition regulière de l'espace à 4 dimensions, I, II”, Izvestia Akad. Nauk. SSSR, Ser. VII (1929), 79110, 147-164.
Delaunay B. N.. “Sur la généralization de la théorie des paralléloèdres”, Izvestia Akad. Nauk SSSR, Ser. VII (1933), 641646.
Fedorov E. S.. Elements of the study of figures (Russian, St. Petersburg, 1885; reprinted, Leningrad, 1953).
Groemer H.. “Ueber Zerlegungen des Euklidischen Raumes”, Math. Z. 79 (1962), 364375.
Groemer H.. “Ueber die Zerlegung des Raumes in homothetische konvexe Körper”, Monatsh. Math. 68 (1964), 2132.
Groemer H.. “Continuity properties of Voronoĭ domains”, Monatsh. Math. 75 (1971), 423431.
Gruber P.. “Geometry of numbers.” In Contributions to geometry ed. Tölke J. and Wills J. M. (Birkhäuser Verlag, 1979), 186225.
Hilbert D.. “Problèmes futurs des mathématiques”, Proc. II Internat. Congr. Math. 1900 (Paris, 1902), 58114.
Kershner R. B.. “On paving the plane”, Amer. Math. Monthly 75 (1968), 839844.
McMullen P.. “Polytopes with centrally symmetric faces”, Israel J. Math. 8 (1970), 194196.
McMullen P.. “Space tiling zonotopes”, Mathematika 22 (1975), 202211.
McMullen P.. “Polytopes with centrally symmetric facets”, Israel J. Math. 23 (1976), 337338.
Minkowski H.. “Allgemeine Lehrsätze über konvexen Polyeder”, Nachr. K. Akad. Wiss. Göttingen, Math.-Phys. Kl. ii (1897), 198219.
Shephard G. C.. “Polytopes with centrally symmetric faces”, Canad. J. Math. 19 (1967), 12061213.
Shephard G. C.. “Space-filling zonotopes”, Mathematika 21 (1974), 261269.
Stein S. K.. “A symmetric star body that tiles but not as a lattice”, Proc. Amer. Math. Soc 36 (1972), 543548.
Voronoĭ G. F.. “Nouvelles applications des paramètres continus a la théorie des formes quadratiques, Deuxième Mémoire I, II”, J. Reine Angew. Math. 134 (1908), 198287; 136 (1909), 67-181.
Žitomirskiĭ O. K.. “Verschärfung eines Satzes von Woronoi”, Z. Leningr. fiz.-mat. Obšč., 2 (1929), 131151.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 89 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd October 2017. This data will be updated every 24 hours.