Structure Results for Multiple Tilings in 3D
Structure Results for Multiple Tilings in 3D
复制标题
3D 中多重平铺的结构结果
DOI：
10.1007/s00454-013-9548-3
复制
发表时间：
2012
影响因子：
0.8
通讯作者：
D. Shiryaev
中科院分区：
文献类型：
--
作者：
N. Gravin;M. N. Kolountzakis;S. Robins;D. Shiryaev
We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $$P$$P is translated with a discrete multiset $$\Lambda $$Λ in such a way that each point of $${\mathbb {R}}^d$$Rd gets covered exactly $$k$$k times, except perhaps the translated copies of the boundary of $$P$$P. It is known that all possible multiple tilers in $${\mathbb {R}}^3$$R3 are zonotopes. In $${\mathbb {R}}^2$$R2 it was known by the work of Kolountzakis (Discrete Comput Geom 23(4):537–553, 2000) that, unless $$P$$P is a parallelogram, the multiset of translation vectors $$\Lambda $$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 $${\mathbb {R}}^3$$R3. Here we prove that this conclusion is indeed true for $${\mathbb {R}}^3$$R3. Namely, we show that if $$P$$P is a convex multiple tiler in $${\mathbb {R}}^3$$R3, with a discrete multiset $$\Lambda $$Λ of translation vectors, then $$\Lambda $$Λ has to be a finite union of translated lattices, unless $$P$$P belongs to a special class of zonotopes. This exceptional class consists of two-flat zonotopes $$P$$P, defined by the Minkowski sum of two 2-dimensional symmetric polygons in $${\mathbb {R}}^3$$R3, 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 $$\Lambda $$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.
We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $$P$$P is translated with a discrete multiset $$\Lambda $$Λ in such a way that each point of $${\mathbb {R}}^d$$Rd gets covered exactly $$k$$k times, except perhaps the translated copies of the boundary of $$P$$P. It is known that all possible multiple tilers in $${\mathbb {R}}^3$$R3 are zonotopes. In $${\mathbb {R}}^2$$R2 it was known by the work of Kolountzakis (Discrete Comput Geom 23(4):537–553, 2000) that, unless $$P$$P is a parallelogram, the multiset of translation vectors $$\Lambda $$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 $${\mathbb {R}}^3$$R3. Here we prove that this conclusion is indeed true for $${\mathbb {R}}^3$$R3. Namely, we show that if $$P$$P is a convex multiple tiler in $${\mathbb {R}}^3$$R3, with a discrete multiset $$\Lambda $$Λ of translation vectors, then $$\Lambda $$Λ has to be a finite union of translated lattices, unless $$P$$P belongs to a special class of zonotopes. This exceptional class consists of two-flat zonotopes $$P$$P, defined by the Minkowski sum of two 2-dimensional symmetric polygons in $${\mathbb {R}}^3$$R3, 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 $$\Lambda $$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.