Rational tilings by $n$-dimensional crosses. II
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- by S. Szabó PDF
- Proc. Amer. Math. Soc. 93 (1985), 569-577 Request permission
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 $i{{\mathbf {e}}_j},\left | i \right | \leq k,1 \leq j \leq n$, 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.References
- W. Hamaker and S. Stein, Splitting groups by integers, Proc. Amer. Math. Soc. 46 (1974), 322–324. MR 349874, DOI 10.1090/S0002-9939-1974-0349874-8
- Sándor Szabó, Rational tilings by $n$-dimensional crosses, Proc. Amer. Math. Soc. 87 (1983), no. 2, 213–222. MR 681824, DOI 10.1090/S0002-9939-1983-0681824-2
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 93 (1985), 569-577
- MSC: Primary 05B45; Secondary 11H31, 20K01, 52A45
- DOI: https://doi.org/10.1090/S0002-9939-1985-0776181-9
- MathSciNet review: 776181