On a Tiling Paradox

A cluster of \( 2n+1 \) cubes comprising the central cube and reflections in all its faces is called the \( n \)-dimensional cube. If \( 2n+1 \) is not a prime, then there are infinitely many tilings of \( \mathbb{R}^n \) by crosses, but it has been conjectured that there is a unique tiling of \( \mathbb{R}^n \) by crosses otherwise. The conjecture has been proved for \( n=2,3 \), and in this paper, we prove it also for \( n=5 \). So there is a unique tiling of \( \mathbb{R}^3 \) by crosses, there are infinitely many tilings of \( \mathbb{R}^4 \), but for \( \mathbb{R}^5 \), there is again only one tiling by crosses. We consider this result to be a paradox as our intuition suggests that “the higher the dimension of the space, the more freedom we get.
“`