Translational tilings by a polytope, with multiplicity

We study the problem of covering Rd by overlapping translates of a convex polytope, such that almost every point of Rd is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which...

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Main Authors: Robins, Sinai., Shiryaev, Dmitry., Gravin, Nick.
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2013
Online Access:https://hdl.handle.net/10356/96779
http://hdl.handle.net/10220/13087
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Institution: Nanyang Technological University
Language: English
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spelling sg-ntu-dr.10356-967792020-03-07T12:31:31Z Translational tilings by a polytope, with multiplicity Robins, Sinai. Shiryaev, Dmitry. Gravin, Nick. School of Physical and Mathematical Sciences We study the problem of covering Rd by overlapping translates of a convex polytope, such that almost every point of Rd is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov [5] and Minkowski [15], and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile Rd. By contrast, for k ≥2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles Rd by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational polytope, is centrally symmetric, and has centrally symmetric facets, then P must k-tile Rd for some positive integer k. 2013-08-12T08:50:31Z 2019-12-06T19:35:01Z 2013-08-12T08:50:31Z 2019-12-06T19:35:01Z 2012 2012 Journal Article Gravin, N., Robins, S.,& Shiryaev, D. (2012). Translational tilings by a polytope, with multiplicity. Combinatorica, 32(6), 629-649. https://hdl.handle.net/10356/96779 http://hdl.handle.net/10220/13087 10.1007/s00493-012-2860-3 en Combinatorica
institution Nanyang Technological University
building NTU Library
country Singapore
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language English
description We study the problem of covering Rd by overlapping translates of a convex polytope, such that almost every point of Rd is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov [5] and Minkowski [15], and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile Rd. By contrast, for k ≥2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles Rd by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational polytope, is centrally symmetric, and has centrally symmetric facets, then P must k-tile Rd for some positive integer k.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Robins, Sinai.
Shiryaev, Dmitry.
Gravin, Nick.
format Article
author Robins, Sinai.
Shiryaev, Dmitry.
Gravin, Nick.
spellingShingle Robins, Sinai.
Shiryaev, Dmitry.
Gravin, Nick.
Translational tilings by a polytope, with multiplicity
author_sort Robins, Sinai.
title Translational tilings by a polytope, with multiplicity
title_short Translational tilings by a polytope, with multiplicity
title_full Translational tilings by a polytope, with multiplicity
title_fullStr Translational tilings by a polytope, with multiplicity
title_full_unstemmed Translational tilings by a polytope, with multiplicity
title_sort translational tilings by a polytope, with multiplicity
publishDate 2013
url https://hdl.handle.net/10356/96779
http://hdl.handle.net/10220/13087
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