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

TL;DR

This paper constructs 4D $A_4$ polytopes and their duals using quaternion representations of the Coxeter-Weyl group, and extends the concept of Catalan solids to higher dimensions.

Contribution

It introduces a quaternion-based method to represent 4D polytopes and their duals, generalizing Catalan solids to four dimensions.

Findings

Constructed 4D $A_4$ polytopes and duals as quaternion orbits.

Projected 4D polytopes into 3D using subgroup $W(A_3)$.

Developed a higher-dimensional generalization of Catalan solids.

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.