Branching of the W(H4) Polytopes and Their Dual Polytopes under the Coxeter Groups W(A4) and W(H3) Represented by Quaternions

TL;DR

This paper explores the structure of 4D H4 polytopes and their duals using quaternion representations of Coxeter groups, analyzing their projections and subgroup branchings.

Contribution

It introduces a quaternion-based method to construct and analyze H4 polytopes and their duals, including their projections and subgroup structures.

Findings

Construction of H4 polytopes and duals via quaternions

Projection preserving icosahedral and tetrahedral subgroups

Branching of polytopes under Coxeter group W(A4)

Abstract

4-dimensional H4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group W(H4) where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary W(H4) orbit into three dimensions is made preserving the icosahedral subgroup W(H3) and the tetrahedral subgroup W(A3), the latter follows a branching under the Coxeter group W(A4) . The dual polytopes of the semi-regular and quasi-regular H4 polytopes have been constructed.