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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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 |
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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. |
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School of Physical and Mathematical Sciences |
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School of Physical and Mathematical Sciences Robins, Sinai. Shiryaev, Dmitry. Gravin, Nick. |
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Robins, Sinai. Shiryaev, Dmitry. Gravin, Nick. |
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Robins, Sinai. Shiryaev, Dmitry. Gravin, Nick. Translational tilings by a polytope, with multiplicity |
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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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