Regular polyhedral groups and reflection groups of rank four

There are three kinds of regular polyhedral groups, the tetrahedral, octahedral and icosahedral groups. Take one of them and write it, say G. Let M be the corresponding regular polyhedra and let p,q,r be the number of vertices of M, that of edges and that of faces, respectively. Then there is a reflection group W of rank four with the condition: the degrees of basic invariants coincide with 2,p,q,r. The first purpose of this paper is to show a relationship among the invariants of W and those of G. The second one is to introduce a system of equations of reflection group W and to give its solutions by reducing it to the homogenization of polyhedral equations for G introduced by Klein.