Characterization of the three-dimensional multiple translative tiles
type: Item
instance of: scholarly article
title: Characterization of the three-dimensional multiple translative tiles
zbMATH Open document ID: 1505.52023
author: Mei Han
author: Kirati Sriamorn
author: Qi Yang
author: Chuan Ming Zong
published in: Advances in Mathematics
publication date: 2022-11-23T00:00:00Z
review text: Let \(X\) be a discrete multiset in the three-dimensional Euclidean space and let \(k\) be a positive integer. \(T+X\) is called a \(k\)-fold translative tiling of \(\mathbb{E}^n\) and \(T\) is called a \(k\)-fold translative tile if every point of \(\mathbb{E}^n\) belongs to at least \(k\) translates of \(T\) in \(T+X\) and belongs to at most \(k\) translates of \(\mathrm{int}(T)\) in \(\mathrm{int}(T)+X\). It is known that any \(k\)-fold translative tile must be a convex polytope. \par The paper proves that in the three-dimensional Euclidean space, every \(k\)-fold translative tile for \(k=2\), 3 and 4 is a parallelohedron, i.e. it must be a parallelotope (parallelepiped), a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron. This statement answers questions of \textit{N. Gravin} et al. [Combinatorica 32, No. 6, 629--648 (2012; Zbl 1299.52019)]. As a comparison, it is known that non-parallelohedral fivefold translative tiles exist in every dimension. Also the structures of a belt of any fivefold and sixfold translative tiles in \(\mathbb{E}^3\) are shown.
reviewed by: Elizaveta Zamorzaeva
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 05B45
zbMATH DE Number: 7622200
zbMATH Keywords: multiple tiling
zbMATH Keywords: zonotope
zbMATH Keywords: parallelohedron
zbMATH Keywords: belt
zbMATH Keywords: dihedral adjacent wheel
MaRDI profile type: Publication
full work available at URL: j.aim.2022.108755
OpenAlex ID: W4307310395
cites work: Q4328176
cites work: Q5787302
cites work: The complete classification of five-dimensional Dirichlet–Voronoi polyhedra of translational lattices
cites work: On the symmetry classification of the four-dimensional parallelohedra
cites work: The Voronoi conjecture for parallelohedra with simply connected \(\delta\)-surfaces
cites work: Translational tilings by a polytope, with multiplicity
cites work: Structure results for multiple tilings in 3D
cites work: Multi-tiling and Riesz bases
cites work: Mathematical problems
cites work: Multiple lattice tiles and Riesz bases of exponentials
cites work: Q3087213
cites work: Multi‐tiling and equidecomposability of polytopes by lattice translates
cites work: Periodic structure of translational multi-tilings in the plane
cites work: Convex bodies which tile space by translation
cites work: Multiple tilings of \(n\)-dimensional space by unit cubes
cites work: On the number of facets of three-dimensional Dirichlet stereohedra. III: Full cubic groups
cites work: Q3036193
cites work: Q4274022
cites work: Regular Dirichlet-Voronoi partitions for the second triclinic group
cites work: Multiple Lattice Tilings in Euclidean Spaces
cites work: Characterization of the Two-Dimensional Fivefold Translative Tiles
cites work: Strange phenomena in convex and discrete geometry. Ed. by James J. Dudziak
cites work: What is known about unit cubes
cites work: Q3374884
cites work: Can You Pave the Plane with Identical Tiles?
DOI: 10.1016/J.AIM.2022.108755
Recommended article: Characterization of the Two-Dimensional Fivefold Translative Tiles
Recommended article: A Local Characterization of Combinatorial Multihedrality in Tilings
Recommended article: Q4328176
Recommended article: Multiple Lattice Tilings in Euclidean Spaces
Recommended article: Multi‐tiling and equidecomposability of polytopes by lattice translates
Recommended article: Q3326358
Recommended article: On tile-\(k\)-transitive tilings of the space
Recommended article: Q3975036
Recommended article: Q5419963
Recommended article: Equitransitive Tilings, or How to Discover New Mathematics
description: scientific article; zbMATH DE number 7622200