Non-periodic Tilings of ℝ n by Crosses

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 ℝ n by crosses have been constructed by several authors for all n∈N. No non-periodic tiling of ℝ n 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 ℝ n 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 ℝ n ,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.