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Convex bodies which tile space by translation

  • P. McMullen (a1)
  • DOI:
  • Published online: 01 February 2010

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.

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S. K. Stein . “A symmetric star body that tiles but not as a lattice”, Proc. Amer. Math. Soc 36 (1972), 543548.

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