Non-periodic Tilings of super( )nby 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 super( )nby crosses have been constructed by several authors for all nN. No non-periodic tiling of super( )nby 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 super( )nby crosses is 2 chi 0 while the total number of periodic Z-tilings is only sub(0). In a sharp contrast to this result we show that any two tilings of super( )nn=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.