  In this paper we construct the quasi regular polyhedra and their duals which
are the generalizations of the Archimedean and Catalan solids respectively.
This work is an extension of two previous papers of ours which were based on
the Archimedean and Catalan solids obtained as the orbits of the Coxeter groups
. When these groups act on an arbitrary vector in 3D Euclidean space they
generate the orbits corresponding to the quasi regular polyhedra. Special
choices of the vectors lead to the platonic and Archimedean solids. In general,
the faces of the quasi regular polyhedra consist of the equilateral triangles,
squares, regular pentagons as well as rectangles, isogonal hexagons, isogonal
octagons, and isogonal decagons depending on the choice of the Coxeter groups
of interest. We follow the quaternionic representation of the group elements of
the Coxeter groups which necessarily leads to the quaternionic representation
of the vertices. We note the fact that the molecule can best be represented by
a truncated icosahedron where the hexagonal faces are not regular, rather, they
are isogonal hexagons where single bonds and double bonds of the carbon atoms
are represented by the alternating edge lengths of isogonal hexagons.
