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Given a fixed collection $U$ of angles, let $\\RU$ be the points that can be obtained by starting with $0$ and $1$, and then recursively adding intersection points of the form $L_\\alpha(p) \\cap L_\\beta(q)$, where $p, q$ have been constructed already, and $\\alpha, \\beta$ are distinct angles in $U$.\n  Our main result is that if $U$ is a group with at least three elem"},"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":"1011.2769","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-11T21:06:35Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"4859f1c3c657555303a624be2f742851fb56ea51981f7c6a77b8f833fa3a8933","abstract_canon_sha256":"48cf2b20876172971e43c774a262ed13f01436db776f46aaa1e4bad8855df5c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:36:08.222075Z","signature_b64":"QE9wZ2ySpN5Z3lOvZ37FYqcmwu616b/CNdf0zmzK0AD5aSG38xFGU+azek9osgjZ0FgjE1meAwAf689WMJ8XAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a139cfa5a04b965563e70d8fb093620e74b7473583520abfcc797125ab2210e6","last_reissued_at":"2026-05-18T04:36:08.221289Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:36:08.221289Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Origami rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Joe Buhler, Ron Graham, Steve Butler, Warwick de Launey","submitted_at":"2010-11-11T21:06:35Z","abstract_excerpt":"Motivated by a question in origami, we consider sets of points in the complex plane constructed in the following way. 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