Abstract
An n-dimensional cross consists of 2n+1 unit cubes: the “central” cube and reflections in all its faces. A tiling by crosses is called a Z-tiling if each cross is centered at a point with integer coordinates. Periodic tilings of ℝn by crosses have been constructed by several authors for all n∈N. No non-periodic tiling of ℝn by crosses has been found so far. We prove that if 2n+1 is not a prime, then the total number of non-periodic Z-tilings of ℝn by crosses is \(2^{\aleph _{0}}\) while the total number of periodic Z-tilings is only ℵ0. In a sharp contrast to this result we show that any two tilings of ℝn,n=2,3, by crosses are congruent. We conjecture that this is the case not only for n=2,3, but for all n where 2n+1 is a prime.
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Horak, P., AlBdaiwi, B. Non-periodic Tilings of ℝn by Crosses. Discrete Comput Geom 47, 1–16 (2012). https://doi.org/10.1007/s00454-011-9373-5
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DOI: https://doi.org/10.1007/s00454-011-9373-5