Rigid Motions of Some Regular Polygons

  • Abba Sani


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 Pn . Our results involve:

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

(2). A new formula, NWTn = n(n-3)/2 for number of ways of triangulating Pn , (the number of ways of cutting Pn 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 Pn as poset and proved some known theorems. 

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


Journal Identifiers

eISSN: 1116-4336