Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes

Emergence of the experimental evidence of E8 in the analysis of one dimensional Ising-model invokes further studies of the Coxeter-Weyl groups generated by reections regarding their applications to polytopes. The Coxeter group W(H₄) which describes the Platonic Polytopes 600-cell and 120-cell in 4D singles out in the mass relations of the bound states of the Ising model for it is a maximal subgroup of the Coxeter-Weyl group W(E8). There exists a one-to-one correspondence between the finite subgroups of quaternions and the Coxeter-Weyl groups of rank 4 which facilitates the study of the rank-4 Coxeter-Weyl groups. In this paper we study the systematic classifications of the 3D-polyhedra and 4D-polytopes through their symmetries described by the rank-3 and rank-4 Coxeter-Weyl groups represented by finite groups of quaternions. We also develop a technique on the constructions of the duals of the polyhedra and the polytopes and give a number of examples. Applications of the rank-2 Coxeter groups have been briey mentioned.