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Harmaala & Kl\\'en [1] provided bounds on P(J) when J is an ellipse or rectangle of eccentricity \\epsilon. We nonrigorously give formulas for P(J) here, in the hope that someone else can fill gaps in our reasoning."},"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":"1608.04299","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-08-12T16:01:14Z","cross_cats_sorted":[],"title_canon_sha256":"5797e825bc0c6b357a34989026e7e1543cf167661850328355dc214ce295f6f6","abstract_canon_sha256":"3964ffe44dd9977d17e9c5b527521a4f80aeea49908fd4fedcfea35df80e5302"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:03.844540Z","signature_b64":"nqh1BlzYdi8ktV5iF+CBbSlfHs9EMVK9XmY6vStV/D0yBUO1m27gc7GSgorDpDPORp0A7eKqt7ZJ53ne12F8Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4882ef4111e2946d4cfd9c76e4112b65d8387c875bac77d038048fa553860851","last_reissued_at":"2026-05-18T01:09:03.844095Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:03.844095Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ptolemy Constants as Described by Eccentricity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Steven Finch","submitted_at":"2016-08-12T16:01:14Z","abstract_excerpt":"Let J denote a simple closed curve in the plane. 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