Non-periodic Tilings of Rn by Crosses

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 a"e (n) by crosses have been constructed by several authors for all naN. No non-periodic tiling of a"e (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 a"e (n) by crosses is 2(N0) while the total number of periodic Z-tilings is only a"mu(0). In a sharp contrast to this result we show that any two tilings of a"e (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.