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Tiling \(\mathbb R^5\) by crosses. (English) Zbl 1297.52006

Summary: 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 \(\mathbb R^n\) by crosses for all \(n\). Al-Bdaiwi 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 \(\mathbb R^{n}\) by crosses, while there is a unique tiling of \(\mathbb R^{n}\) 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 \(\mathbb R^{5}\) there is a unique regular, and no non-regular, tiling by crosses. So there is a unique tiling of \(\mathbb R^{3}\) by crosses, there are \(2^{\aleph_{0}}\) tilings of \(\mathbb R^{4}\), but for \(\mathbb R^{5}\) 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’.

MSC:

52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
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