4D-Polytopes and Their Dual Polytopes of the Coxeter Group $W(A_{4})$ Represented by Quaternions

Abstract

4-dimensional $A_{4}$ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group $W(A_{4})$ where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary $W(A_{4})$ orbit into three dimensions is made using the subgroup $W(A_{3})$. A generalization of the Catalan solids for 3D polyhedra has been developed and dual polytopes of the uniform $A_{4}$ polytopes have been constructed.