{"paper":{"title":"Spatial chaos of Wang tiles with two symbols","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Jin-Yu Chen, Song-Sun Lin, Wen-Guei Hu, Yu-Jie Chen","submitted_at":"2015-07-15T02:17:26Z","abstract_excerpt":"This investigation completely classifies the spatial chaos problem in plane edge coloring (Wang tiles) with two symbols. For a set of Wang tiles $\\mathcal{B}$, spatial chaos occurs when the spatial entropy $h(\\mathcal{B})$ is positive. $\\mathcal{B}$ is called a minimal cycle generator if $\\mathcal{P}(\\mathcal{B})\\neq\\emptyset$ and $\\mathcal{P}(\\mathcal{B}')=\\emptyset$ whenever $\\mathcal{B}'\\subsetneqq \\mathcal{B}$, where $\\mathcal{P}(\\mathcal{B})$ is the set of all periodic patterns on $\\mathbb{Z}^{2}$ generated by $\\mathcal{B}$. Given a set of Wang tiles $\\mathcal{B}$, write $\\mathcal{B}=C_{1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04070","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":""},"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"}