Rigid Motions of Some Regular Polygons

Abstract

We examined permutations of vertices/sides of some regular shapes viewed as rigid motions. In particular, we use combinatorial techniques to enumerate symmetric permutations of vertices/ sides of an n-sided regular polygon P_n . Our results involve:

(1) A well knownformula, NSYP_n = 2n for generating the number of symmetries in an n-sided regular polygon accomplished using permutations;

(2). A new formula, NWT_n = ^n(n-3)/₂ for number of ways of triangulating P_n , (the number of ways of cutting P_n into triangles by connecting its vertices with straight lines); thereby providing a proof for Richard and Stanley’s conjecture that “All diagonals are flipped in a geodesics between two antipodes in exactly ^n(n-3)/₂ ”.We also examined the set S = [n] = {1,2,...n) of vertices of P_n as poset and proved some known theorems. 

A discussion is given of lattices whose maximum length chains correspond to restricted permutations.