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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.2229","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-06-05T01:28:00Z","cross_cats_sorted":[],"title_canon_sha256":"fc7d5f02a3796b0b7f356279680ce808e5e25586d43baf8722af478208246d8c","abstract_canon_sha256":"626ecbf3e94bf656ad9fff81ebb712352339e846875469e18fe6cf8b8fa32653"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:07.949955Z","signature_b64":"doskAwLZIgm6JT1Zu+wdACHK5p557LHMYaIXCH+MCa+1DgGtGG4iJ8+IlYESkTA5VBBPUQyLVDiTS6sdoursAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f33b509f0f79b19e76fb788ec0bc2c60e01eef1e3d9c103602c4430437db9efe","last_reissued_at":"2026-05-17T23:54:07.949480Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:07.949480Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $(2\\alpha, \\alpha, 2\\beta)$. 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