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Space tiling zonotopes

Published online by Cambridge University Press:  26 February 2010

P. McMullen
P. McMullen
Affiliation:
University College London, Gower Street, London WC1E 6BT.
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Abstract

A d-dimensional zonotope Z in Ed which is the vector sum of n line segments is linearly equivalent to the image of a regular n-cube under some orthogonal projection. The zonotope in En-d which is the image of the same cube under projection on to the orthogonal complementary subspace is said to be associated with Z. In this paper is proved a conjecture of G. C. Shephard, which asserts that, if Z tiles Ed by translation, with adjacent zonotopes meeting facet against facet, then tiles En-d in the same manner. A number of conditions, conjectured by Shephard and H. S. M. Coxeter to be equivalent to the tiling property, are also proved.

Type
Research Article
Information
Mathematika , Volume 22 , Issue 2 , December 1975 , pp. 202 - 211
Copyright
Copyright © University College London 1975

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References

Coxeter, H. S. M.. “The classification of zonohedra by means of projective diagrams”, J. Math. Pures Appi, 41 (1962), 137–156; MR25#4417.Google Scholar
McMullen, P.. “On zonotopes”, Trans. Amer. Math. Soc., 159 (1971), 91–110; MR43#5410.CrossRefGoogle Scholar
Shephard, G. C.. “Combinatorial properties of associated zonotopes”, Canad. J. Math., 26 (1974), 302–321.CrossRefGoogle Scholar
Shephard, G. C.. “Space filling zonotopes”, Mathematika, 21 (1974), 261–269.Google Scholar