Structure results for multiple tilings in 3D

TL;DR

This paper proves that in 3D, convex multiple tilers generally have translation vectors forming a finite union of lattices, with exceptions involving special zonotopes that can admit aperiodic tilings, extending 2D results to 3D.

Contribution

It establishes that 3D multiple tilings by convex bodies mostly involve quasi-periodic translation sets, except for a class of zonotopes allowing aperiodic tilings, confirming a 2002 conjecture in higher dimensions.

Findings

Most 3D multiple tilers have translation vectors as finite unions of lattices.

Special two-flat zonotopes can admit aperiodic multiple tilings.

Fourier methods were used to discover non-quasi-periodic tilings.

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.