Tiling R 5 by Crosses

Abstract

An n-dimensional cross comprises 2n+1 unit cubes: the center cube and reflections in all its faces. It is well known that there is a tiling of Rn by crosses for all n. AlBdaiwi and the first author proved that if 2n+1 is not a prime then there are $2^{\aleph_{0}}$ non-congruent regular (= face-to-face) tilings of Rn by crosses, while there is a unique tiling of Rn by crosses for n=2,3. They conjectured that this is always the case if 2n+1 is a prime. To support the conjecture we prove in this paper that also for R5 there is a unique regular, and no non-regular, tiling by crosses. So there is a unique tiling of R3 by crosses, there are $2^{\aleph_{0}}$ tilings of R4, but for R5 there is again only one tiling by crosses. We guess that this result goes against our intuition that suggests `the higher the dimension of the space, the more freedom we get'.