Structure Results for Multiple Tilings in 3D

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 ℝ^d gets covered exactly k times, except perhaps the translated copies of the boundary of P . It is known that all possible multiple tilers in ℝ^3 are zonotopes. In ℝ^2 it was known by the work of Kolountzakis (Discrete Comput Geom 23(4):537–553, 2000 ) 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 (Kolountzakis, Discrete Comput Geom 23(4):537–553, 2000 ) the author asked whether the same quasi-periodic structure on the translation vectors would be true in ℝ^3 . Here we prove that this conclusion is indeed true for ℝ^3 . Namely, we show that if P is a convex multiple tiler in ℝ^3 , 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 ℝ^3 , 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 (nonquasi-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.