Abstract
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 \(\lambda \ge 0\) 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 the polygon is affinely regular, but in some special cases there are polygons which satisfy this property but are not affinely regular. The proofs are based on the use of linear algebraic and number theoretic tools. In addition, we apply our method to characterize polytopes with certain symmetry groups.
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Acknowledgements
The author expresses his gratitude to M. Naszódi and K. Swanepoel for the fruitful discussions they had on this subject, to K. Swanepoel for directing his attention to the results in [1], and to an anonymous referee for his/her helpful comments, in particular for giving an idea to prove (3.3) in a different way.
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Partially supported by the National Research, Development and Innovation Office, NKFI-119670.
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Lángi, Z. A characterization of affinely regular polygons. Aequat. Math. 92, 1037–1049 (2018). https://doi.org/10.1007/s00010-018-0541-z
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DOI: https://doi.org/10.1007/s00010-018-0541-z