ISSN 1088-6826(online) ISSN 0002-9939(print)

Rational tilings by -dimensional crosses

Author: Sándor Szabó
Journal: Proc. Amer. Math. Soc. 87 (1983), 213-222
MSC: Primary 05B45; Secondary 10E30, 20K01, 52A45
DOI: https://doi.org/10.1090/S0002-9939-1983-0681824-2
MathSciNet review: 681824
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Abstract: Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors and whose centers are , , in -dimensional Euclidean space. The union of these cubes is called a cross. This cross consists of cubes; a central cube together with arms of length . A family of translates of a cross whose union is -dimensional Euclidean space and whose interiors are disjoint is a tiling. Denote the set of translation vectors by . If the vector set is a vector lattice, then we say that the tiling is a lattice tiling. If every vector of has rational coordinates, then we say that the tiling is a rational tiling, and, similarly, if every vector of has integer coordinates, then we say that the tiling is an integer tiling. Is there a noninteger tiling by crosses? In this paper we shall prove that if there is an integer lattice tiling by crosses, if is not a prime, and if for every prime divisor of , then there is a rational noninteger lattice tiling by crosses and there is an integer nonlattice tiling by crosses. We will illustrate this in the case of a cross with arms of length 2 in -dimensional Euclidean space. Throughout, the techniques are algebraic.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0681824-2
Keywords: Exact sequence, factorization of abelian groups, lattices, tiling, star body
Article copyright: © Copyright 1983 American Mathematical Society