{"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 $m_1,m_2,k \\in \\mathbb{Z}$. In particular, we show that in `most' cases this implies that t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03036","kind":"arxiv","version":2},"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"}