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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Rational tilings by $ n$-dimensional crosses


Author: Sándor Szabó
Journal: Proc. Amer. Math. Soc. 87 (1983), 213-222
MSC: Primary 05B45; Secondary 10E30, 20K01, 52A45
MathSciNet review: 681824
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Abstract: Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors $ {{\mathbf{e}}_1}, \ldots ,{{\mathbf{e}}_n}$ and whose centers are $ i{{\mathbf{e}}_j}$, $ 0 \leqslant \vert i\vert \leqslant k$, in $ n$-dimensional Euclidean space. The union of these cubes is called a cross. This cross consists of $ 2kn + 1$ cubes; a central cube together with $ 2n$ arms of length $ k$. A family of translates of a cross whose union is $ n$-dimensional Euclidean space and whose interiors are disjoint is a tiling. Denote the set of translation vectors by $ {\mathbf{L}}$. If the vector set $ {\mathbf{L}}$ is a vector lattice, then we say that the tiling is a lattice tiling. If every vector of $ {\mathbf{L}}$ has rational coordinates, then we say that the tiling is a rational tiling, and, similarly, if every vector of $ {\mathbf{L}}$ 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 $ 2kn + 1$ is not a prime, and if $ p > k$ for every prime divisor $ p$ of $ 2kn + 1$, 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 $ 55$-dimensional Euclidean space. Throughout, the techniques are algebraic.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0681824-2
PII: S 0002-9939(1983)0681824-2
Keywords: Exact sequence, factorization of abelian groups, lattices, tiling, star body
Article copyright: © Copyright 1983 American Mathematical Society