A characterization of affinely regular polygons

Abstract

We characterize affinely regular polygons in a real affine space as follows. Consider two polygons P and Q with the same number of vertices and a common side, P being an affine image of some (convex or nonconvex) regular polygon. Q is then an affine image of the same regular polygon if and only if the vectors between the corresponding vertices of P and Q are uniquely determined multiples of the vector between the vertices just after the common side.

How Bürgi computed the sines of all integer angles simultaneously in 1586

“…The use of discrete Fourier transforms in the geometry of polygons goes back to Darboux in 1878, who studied the iterated polygons constructed on the side midpoints [5]. This very powerful tool often leads to one-line proofs [1,7,10,17,18,19,20,21,22,25,26,27,29] (more references can be found in [17]).…”

A characterization of affinely regular polygons

“…They appear, for example, in geometry as extremal sets in optimization problems [10] or in the famous Napoleon-Barlotti Theorem, redisovered by Gerber in 1980 [9], or in discrete tomography [8]; in linear algebra as sets of vectors cyclically permuted by unimodular matrices. In the 1970s, Coxeter started a systematic investigation of the properties of affinely regular polygons [2] in the Euclidean plane, which was later continued both in the Euclidean plane [3], [13], [17], and in affine planes in general [4], [15], [16]. For a survey about these properties, particularly about characterizations of affinely regular polygons, the interested reader is referred to [7].…”