{"paper":{"title":"A self-similar aperiodic set of 19 Wang tiles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"S\\'ebastien Labb\\'e","submitted_at":"2018-02-09T14:11:13Z","abstract_excerpt":"We define a Wang tile set $\\mathcal{U}$ of cardinality 19 and show that the set $\\Omega_\\mathcal{U}$ of all valid Wang tilings $\\mathbb{Z}^2\\to\\mathcal{U}$ is self-similar, aperiodic and is a minimal subshift of $\\mathcal{U}^{\\mathbb{Z}^2}$. Thus $\\mathcal{U}$ is the second smallest self-similar aperiodic Wang tile set known after Ammann's set of 16 Wang tiles. The proof is based on the unique composition property. We prove the existence of an expansive, primitive and recognizable $2$-dimensional morphism $\\omega:\\Omega_\\mathcal{U}\\to\\Omega_\\mathcal{U}$ that is onto up to a shift. The proof of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.03265","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}