Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions II

TL;DR

This paper extends the construction of quasi regular polyhedra and their duals using Coxeter groups and quaternionic representations, generalizing classical solids and exploring their geometric and chemical implications.

Contribution

It introduces a quaternionic approach to generate quasi regular polyhedra and their duals, broadening the understanding of polyhedral symmetries beyond classical models.

Findings

Generated quasi regular polyhedra with diverse face types

Connected polyhedral structures to molecular models like truncated icosahedra

Demonstrated quaternionic representation of Coxeter group elements

Abstract

In this paper we construct the quasi regular polyhedra and their duals which are the generalizations of the Archimedean and Catalan solids respectively. This work is an extension of two previous papers of ours which were based on the Archimedean and Catalan solids obtained as the orbits of the Coxeter groups . When these groups act on an arbitrary vector in 3D Euclidean space they generate the orbits corresponding to the quasi regular polyhedra. Special choices of the vectors lead to the platonic and Archimedean solids. In general, the faces of the quasi regular polyhedra consist of the equilateral triangles, squares, regular pentagons as well as rectangles, isogonal hexagons, isogonal octagons, and isogonal decagons depending on the choice of the Coxeter groups of interest. We follow the quaternionic representation of the group elements of the Coxeter groups which necessarily leads to the quaternionic representation of the vertices. We note the fact that the molecule can best be represented by a truncated icosahedron where the hexagonal faces are not regular, rather, they are isogonal hexagons where single bonds and double bonds of the carbon atoms are represented by the alternating edge lengths of isogonal hexagons.