Formulas Among Diagonals in the Regular Polygon and the Catalan Numbers

We look at relationships among the lengths of diagonals in the regular polygon. Specifically, we derive formulas for all diagonals in terms of the shortest diagonals and other formulas in terms of the next-to-shortest diagonals, assuming unit side length. These formulas are independent of the number of sides of the regular polygon. We also show that the formulas in terms of the shortest diagonals involve the famous Catalan numbers. 1. Motivation In [1], Fontaine and Hurley develop formulas that relate the diagonal lengths of a regular n-gon. Specifically, given a regular convex n-gon whose vertices are P0, P1, . . . , Pn−1, define dk as the distance between P0 and Pk. Then the law of sines yields dk dj = sin kπ n sin jπ n . Defining rk = sin kπ n sin πn , the formulas given in [1] are rhrk = min{k,h,n−k,n−h} ∑ i=1 r|k−h|+2i−1 and 1 rk = s ∑ j=1 rk(2j−1) where s = min{j > 0 : jk ≡ ±1 mod n}. Notice that for 1 ≤ k ≤ n − 1, rk = dk d1 , but there is no a priori restriction on k in the definition of rk. Thus, it would make perfect sense to consider r0 = 0 and r−k = −rk not to mention rk for non-integer values of k as well. Also, the only restriction on n in the definition of rn is that n not be zero or the reciprocal of an integer. Publication Date: October 23, 2006. Communicating Editor: Paul Yiu.