Tiling $R^{5}$ by Crosses

TL;DR

This paper proves that in five-dimensional space, there is a unique regular tiling of ^{5} by crosses, supporting a conjecture that prime-related dimensions have unique tilings, contrasting with non-prime cases where many tilings exist.

Contribution

The paper confirms the conjecture that ^{5} has a unique regular tiling by crosses, extending previous results and showing a pattern related to the primality of 2n+1.

Findings

Unique tiling of ^{5} by crosses

Multiple tilings in ^{4}

Single tiling in ^{3}

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 $R^{n}$ 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 $R^{n}$ by crosses, while there is a unique tiling of $R^{n}$ 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 $R^{5}$ there is a unique regular, and no non-regular, tiling by crosses. So there is a unique tiling of $R^{3}$ by crosses, there are $2^{\aleph_{0}}$ tilings of $R^{4},$ but for $R^{5}$ 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".