الملخص الإنجليزي
3-dimensional convex uniform polyhedra can be obtained from Coxeter groups W(A3), W(B3) and W (Hz) which describe the tetrahedral, octahedral and icosahedral symmetry groups, respectively. The vertices of these polyhedra are the orbits of the Coxeter groups when they act on the highest weights of the Coxeter-Dynkin diagrams. In this thesis, we develop a technique to describe the configuration of Platonic, Archimedean and Catalan solids by determining the number of their vertices, edges, and faces. The orthogonal projection of the convex uniform polyhedra on the Coxeter plane which is defined by the simple roots of the Coxeter diagram 12(h) is studied. The projected vertices of a polyhedron onto the Coxeter plane constitute an orbit of the dihedral symmetry Dn. In this thesis, we use the quaternionic representations of the Coxeter group elements.
