A symmetric star polyhedron that tiles but not as a fundamental domain

Szabó, Sándor

doi:10.1090/S0002-9939-1983-0715888-4

A symmetric star polyhedron that tiles but not as a fundamental domain
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Proc. Amer. Math. Soc. 89 (1983), 563-566 Request permission

Abstract:

In [7] S. K. Stein constructed a 10-dimensional centrally-symmetric star body whose translates tile $10$-space but whose translates by a lattice do not tile it. In [8] he constructed a $5$-dimensional star polyhedron whose translates tile $5$-space but whose congruent copies by a group of motions do not tile it. So there is no lattice tiling by translates of this polyhedron. In the present paper we shall construct a $5$-dimensional centrally-symmetric star polyhedron whose translates tile $5$-space but whose congruent copies by a group of motions do not tile it. Furthermore, this phenomenon occurs at an infinitude of dimensions.

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