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


Author: Sándor Szabó
Journal: Proc. Amer. Math. Soc. 89 (1983), 563-566
MSC: Primary 05B45; Secondary 52A45
DOI: https://doi.org/10.1090/S0002-9939-1983-0715888-4
MathSciNet review: 715888
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0715888-4
Keywords: Tiling, star body, factorization of finite abelian groups, groups of motions, lattice
Article copyright: © Copyright 1983 American Mathematical Society

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