Arithmetic Polygons

We consider the question of the existence of equiangular polygons with edge lengths in arithmetic progression, and show that they do not exist when the number of sides is a power of two and do exist if it is any other even number. A few results for small odd numbers are given. An Olympiad problem [1] asked the following. Prove that there exists a convex 1990-gon with the following two properties: (a) all angles are equal. (b) the lengths of the 1990 sides are the numbers 12, 22, 32, . . . , 19902 in some order. Recently, remembering the question imperfectly, I constructed a convex 1990-gon with side lengths 1, 2, 3, . . . , 1990, and wondered for which other N such an N -gon could be constructed. This note gives a partial answer. Define an arithmetic polygon to be an equiangular polygon with edge lengths forming (upon suitable rearrangement) a nondegenerate arithmetic sequence. Lemma 1. For any N, if there exists an arithmetic N-gon, there exists one such with edge lengths 1, 2, . . . , N. Proof. We work in the complex plane, so that an edge of length L oriented at an angle θ to the positive real axis is represented by the complex number Leiθ . An equiangular N -gon has edge orientations (in cyclic order) e2( j/N )π i , and edge lengths a + p( j)b for some permutation p : (0, 1, . . . , n − 1)→ (0, 1, . . . , n − 1). This polygonal path closes if and only if