{"paper":{"title":"An aperiodic set of Wang tiles for every quadratic irrational","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.DS","math.MG"],"primary_cat":"math.NT","authors_text":"Jarkko Kari, Pieter Mostert, S\\'ebastien Labb\\'e","submitted_at":"2026-06-23T15:22:15Z","abstract_excerpt":"We propose a sufficient condition for the non-periodicity of a set of Wang tiles. It applies to sets of Wang tiles whose tiles have vertical or horizontal stripes. The proof is based on a geometric argument involving a quadrilateral circumscribed to a parabola from which we conclude the irrationality of the densities of the vertical and horizontal stripes. We apply the sufficient condition to propose new proofs of non-periodicity of known sets of Wang tiles, including an encoding of Penrose tilings into 24 Wang tiles and the family of metallic mean Wang tiles.\n  Conversely, for every pair $(\\a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24693","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.24693/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}