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- W4296577393 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 $Lambda$ in such a way that each point of the space gets covered exactly $k$ times, except perhaps the translated copies of the boundary of $P$. It is known that all possible multiple tilers in 3D are zonotopes. In 2D it was known by the work of M. Kolountzakis that, unless $P$ is a parallelogram, the multiset of translation vectors $Lambda$ must be a finite union of translated lattices (also known as quasi periodic sets). In that work [Kolountzakis, 2002], the author asked whether the same quasi-periodic structure on the translation vectors would be true in 3D. Here we prove that this conclusion is indeed true for 3D. Namely, we show that if $P$ is a convex multiple tiler in 3D, with a discrete multiset $Lambda$ of translation vectors, then $Lambda$ 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 $n+m$ line segments that lie in the union of two different two-dimensional subspaces $H_1$ and $H_2$. Equivalently, a two-flat zonotope $P$ may be thought of as the Minkowski sum of two 2-dimensional symmetric polygons one of which may degenerate into a single line segment. It turns out that rational two-flat zonotopes admit a multiple tiling with an aperiodic (non-quasi-periodic) set of translation vectors $Lambda$. 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." @default.
- W4296577393 created "2022-09-22" @default.
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- W4296577393 date "2012-08-07" @default.
- W4296577393 modified "2023-09-29" @default.
- W4296577393 title "Structure results for multiple tilings in 3D" @default.
- W4296577393 doi "https://doi.org/10.48550/arxiv.1208.1439" @default.
- W4296577393 hasPublicationYear "2012" @default.
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