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Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile
We construct an example of a group [Formula: see text] for a finite abelian group [Formula: see text] , a subset E of [Formula: see text] , and two finite subsets [Formula: see text] of G, such that it is undecidable in ZFC whether [Formula: see text] can be tiled by translations of [Formula: see te...
Autores principales: | , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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Springer US
2023
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10676348/ https://www.ncbi.nlm.nih.gov/pubmed/38022896 http://dx.doi.org/10.1007/s00454-022-00426-4 |
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author | Greenfeld, Rachel Tao, Terence |
author_facet | Greenfeld, Rachel Tao, Terence |
author_sort | Greenfeld, Rachel |
collection | PubMed |
description | We construct an example of a group [Formula: see text] for a finite abelian group [Formula: see text] , a subset E of [Formula: see text] , and two finite subsets [Formula: see text] of G, such that it is undecidable in ZFC whether [Formula: see text] can be tiled by translations of [Formula: see text] . In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles [Formula: see text] , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in [Formula: see text] ). A similar construction also applies for [Formula: see text] for sufficiently large d. If one allows the group [Formula: see text] to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. |
format | Online Article Text |
id | pubmed-10676348 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2023 |
publisher | Springer US |
record_format | MEDLINE/PubMed |
spelling | pubmed-106763482023-01-04 Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile Greenfeld, Rachel Tao, Terence Discrete Comput Geom Article We construct an example of a group [Formula: see text] for a finite abelian group [Formula: see text] , a subset E of [Formula: see text] , and two finite subsets [Formula: see text] of G, such that it is undecidable in ZFC whether [Formula: see text] can be tiled by translations of [Formula: see text] . In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles [Formula: see text] , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in [Formula: see text] ). A similar construction also applies for [Formula: see text] for sufficiently large d. If one allows the group [Formula: see text] to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. Springer US 2023-01-04 2023 /pmc/articles/PMC10676348/ /pubmed/38022896 http://dx.doi.org/10.1007/s00454-022-00426-4 Text en © The Author(s) 2023 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Article Greenfeld, Rachel Tao, Terence Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile |
title | Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile |
title_full | Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile |
title_fullStr | Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile |
title_full_unstemmed | Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile |
title_short | Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile |
title_sort | undecidable translational tilings with only two tiles, or one nonabelian tile |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10676348/ https://www.ncbi.nlm.nih.gov/pubmed/38022896 http://dx.doi.org/10.1007/s00454-022-00426-4 |
work_keys_str_mv | AT greenfeldrachel undecidabletranslationaltilingswithonlytwotilesoronenonabeliantile AT taoterence undecidabletranslationaltilingswithonlytwotilesoronenonabeliantile |