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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. II


Author: S. Szabó
Journal: Proc. Amer. Math. Soc. 93 (1985), 569-577
MSC: Primary 05B45; Secondary 11H31, 20K01, 52A45
MathSciNet review: 776181
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Abstract: The union of translates of a closed unit $ n$-dimensional cube whose edges are parallel to the coordinate unit vectors $ {{\mathbf{e}}_1}, \ldots ,{{\mathbf{e}}_n}$ and whose centers are $ i{{\mathbf{e}}_j},\left\vert i \right\vert \leq k,1 \leq j \leq n$, is called a $ (k,n)$-cross. A system of translates of a $ (k,n)$-cross is called an integer (a rational) lattice tiling if its union is $ n$-space and the interiors of its elements are disjoint, the translates form a lattice and each translation vector of the lattice has integer (rational) coordinates. In this paper we shall continue the examination of rational cross tilings begun in [2], constructing rational lattice tilings by crosses that have noninteger coordinates on several axes.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0776181-9
PII: S 0002-9939(1985)0776181-9
Keywords: Factorization splitting and partition of abelian groups, crosses, semicrosses, lattice tiling
Article copyright: © Copyright 1985 American Mathematical Society