Branching of the W($H_4$) polytopes and their dual polytopes under the coxeter groups W($A_4$) and W($H_3$) represented by quaternions

Abstract

4-dimensional H4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter- Weyl group W(($H_4$) , where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary $W(H_4)$ orbit into three dimensions is made preserving the icosahedral subgroup W(($H_3$) and the tetrahedral subgroup W($A_3$) . The latter follows a branching under the Coxeter group W($A_4$) . The dual polytopes of the semi-regular and quasi-regular $H_4$ polytopes have been constructed.