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The aim of this paper is to examine the properties of polygons whose vertices $p_1,p_2,\\ldots,p_n \\in \\mathbb{C}$ satisfy the property that $p_{j+m_1}-p_{j+m_2} = w (p_{j+k}-p_j)$ for some $w \\in \\mathbb{C}$ and $m_1,m_2,k \\in \\mathbb{Z}$. In particular, we show that in `most' cases this implies that t"},"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":"1706.03036","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-06-09T16:49:55Z","cross_cats_sorted":["math.FA","math.NT"],"title_canon_sha256":"e965e18a21ca145c8c3969ba96f774743f25be3b89f5c402a1d117c8cfd6b5a6","abstract_canon_sha256":"a018a0da425eb23619c5278cd41f9571064157a0fd2c33b5bb15089a151b69a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:42.043266Z","signature_b64":"2OhrI0qcGbDRraxkJMjdx03c/GXTwZnvTL9E8INUOZI5d9OTiaZEBPj/15krrC2ezCPxzF8ZldUNAANuKzxWCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85b034aeb1125b33773a055106a8b03da10896c8ecad055ad1fb49653bb5855c","last_reissued_at":"2026-05-18T00:25:42.042756Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:42.042756Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A characterization of affinely regular polygons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.NT"],"primary_cat":"math.MG","authors_text":"Zsolt Langi","submitted_at":"2017-06-09T16:49:55Z","abstract_excerpt":"In 1970, Coxeter gave a short and elegant geometric proof showing that if $p_1, p_2, \\ldots, p_n$ are vertices of an $n$-gon $P$ in cyclic order, then $P$ is affinely regular if, and only if there is some $\\lambda \\geq 0$ such that $p_{j+2}-p_{j-1} = \\lambda (p_{j+1}-p_j)$ for $j=1,2,\\ldots, n$. The aim of this paper is to examine the properties of polygons whose vertices $p_1,p_2,\\ldots,p_n \\in \\mathbb{C}$ satisfy the property that $p_{j+m_1}-p_{j+m_2} = w (p_{j+k}-p_j)$ for some $w \\in \\mathbb{C}$ and $m_1,m_2,k \\in \\mathbb{Z}$. 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