Abstract
We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body P is translated with a discrete multiset Λ in such a way that each point of ℝd gets covered exactly k times, except perhaps the translated copies of the boundary of P. It is known that all possible multiple tilers in ℝ3 are zonotopes. In ℝ2 it was known by the work of M. Kolountzakis [9] that, unless P is a parallelogram, the multiset of translation vectors Λ must be a finite union of translated lattices (also known as quasi periodic sets). In that work [9] the author asked whether the same quasi-periodic structure on the translation vectors would be true in ℝ3. Here we prove that this conclusion is indeed true for ℝ3.
Namely, we show that if P is a convex multiple tiler in ℝ3, with a discrete multiset Λ of translation vectors, then Λ has to be a finite union of translated lattices, unless P belongs to a special class of zonotopes. This exceptional class consists of two-flat zonotopes P, defined by the Minkowski sum of two 2-dimensional symmetric polygons in ℝ3, one of which may degenerate into a single Une segment. It turns out that rational two-flat zonotopes admit a multiple tiling with an aperiodic (non-quasi-periodic) set of translation vectors Λ. We note that it may be quite difficult to offer a visualization of these 3-dimensional non-quasi-periodic tilings, and that we discovered them by using Fourier methods.
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Gravin, N., Kolountzakis, M.N., Robins, S., Shiryaev, D. (2013). Extended abstract for structure results for multiple tilings in 3D. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_30
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DOI: https://doi.org/10.1007/978-88-7642-475-5_30
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