{"paper":{"title":"Triangle Tiling: The case $3\\alpha + 2\\beta = \\pi$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Michael Beeson","submitted_at":"2012-06-05T01:28:00Z","abstract_excerpt":"An $N$-tiling of triangle $ABC$ by triangle $T$ (the `tile') is a way of writing $ABC$ as a union of $N$ copies of $T$ overlapping only at their boundaries. Let the tile $T$ have angles $(\\alpha,\\beta,\\gamma)$, and sides $(a,b,c)$. This paper takes up the case when $3\\alpha + 2\\beta = \\pi$. Then there are (as was already known) exactly five possible shapes of $ABC$: either $ABC$ is isosceles with base angles $\\alpha$, $\\beta$, or $\\alpha+\\beta$, or the angles of $ABC$ are $(2\\alpha,\\beta,\\alpha+\\beta)$, or the angles of $ABC$ are $(2\\alpha, \\alpha, 2\\beta)$. In each of these cases, we have dis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2229","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"}