On a Tiling Paradox

Abstract

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.
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