Hypercube related polytopes

Abstract

A regular polyhedron is a polyhedron having congruent regular polygons as faces, arranged in the same manner around identical vertices; its symmetry group acts transitively on its flags, a regular polyhedron being vertex-, edgeand face-transitive [1]. They show three symmetry groups: tetrahedral; octahedral (or cubic) and icosahedral (or dodecahedral). Any shapes with icosahedral or octahedral symmetry will also include the tetrahedral symmetry. There are five regular polyhedra, known as Platonic polyhedral solids: tetrahedron (T), cube (C), octahedron (O), dodecahedron (D) and icosahedron (I), written as {3,3}; {4,3}; {3,4}; {5,3} and {3,5} by using the basic Schlӓfli [2] symbols {p,q} where p is the number of vertices in a given face while q is the number of faces containing a given vertex.