{"title":"Tiling with Three Polygons is Undecidable","authors":"Erik D. Demaine, Stefan Langerman","doi":"arxiv-2409.11582","DOIUrl":null,"url":null,"abstract":"We prove that the following problem is co-RE-complete and thus undecidable:\ngiven three simple polygons, is there a tiling of the plane where every tile is\nan isometry of one of the three polygons (either allowing or forbidding\nreflections)? This result improves on the best previous construction which\nrequires five polygons.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"17 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11582","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that the following problem is co-RE-complete and thus undecidable:
given three simple polygons, is there a tiling of the plane where every tile is
an isometry of one of the three polygons (either allowing or forbidding
reflections)? This result improves on the best previous construction which
requires five polygons.
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