All regular and semi-regular polyhedra can be obtained from Coxeter groups W (A3), W(Bz) and W(Hz) which are the tetrahedral, octahedral and icosahedral symmetry groups respectively. In this thesis, the Platonic and Archimedean solids were constructed as the orbits of the Coxeter groups acting on the highest weights of Coxeter-Dynkin diagrams. Duals of the Archimedean solids so called the Catalan solids were obtained, up to some scale factors, as the union of orbits of the fundamental weights of Coxeter groups. There are two chiral Archimedean solids, the snub cube and the snub dodecahedron that were constructed from the proper rotational subgroups of W(B3) and W(H3) respectively. Their duals, the Catalan solids, pentagonal icositetrahedron and pentagonal hexecontahedron were obtained as the unions of three orbits of the corresponding groups. All the regular and semi-regular polyhedra were constructed in the context of the quaternionic representations of the root systems of the Coxeter groups.