Counting Crossings Among N Regular Polygons Inscribed in a Circle
Abstract
Consider regular polygons P_3, P_4, ..., P_N inscribed in a circle and aligned so that they share a common vertex. Counting the distinct proper crossings among their sides as N increases produces an integer sequence. We show that any two polygons P_m and P_k, with 3 <= m < k, have exactly 2(m - gcd(m,k)) proper side crossings. We further prove that no proper crossing is shared by sides from three or more polygons of distinct orders. It follows that for every integer N >= 3, the total number of distinct proper crossing points is exactly A(N) = 2 * Sum_{3 <= m < k <= N} (m - gcd(m,k)). We also give an arithmetic reformulation of the sequence in terms of Pillai's arithmetical function. Finally, we consider a second polygon alignment rule in which the midpoint of one side of every polygon lies on a common fixed radius, yielding a second integer sequence of crossing numbers.Certificates
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