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

TL;DR

This paper explores quasi regular polygons and polyhedra with Coxeter symmetries using quaternions, presenting new aperiodic tilings and potential applications to graphene structures.

Contribution

It introduces a quaternion-based method for constructing quasi regular polygons and polyhedra, including aperiodic tilings and duals, extending Coxeter symmetry applications.

Findings

Derived isogonal hexagons, octagons, and decagons from Coxeter diagrams.

Presented aperiodic tilings of the plane with regular and isogonal polygons.

Proposed a model of graphene with a specific aperiodic tiling pattern.

Abstract

In two series of papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper we discuss the quasi regular polygons (isogonal and isotoxal polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain aperiodic tilings of the plane with the isogonal polygons along with the regular polygons. We point out that one type of aperiodic tiling of the plane with regular and isogonal hexagons may represent a state of graphene where one carbon atom is bound to three neighboring carbons with two single bonds and one double bond. We also show how the plane can be tiled with two tiles; one of them is the isotoxal polygon, dual of the isogonal polygon. A general method is employed for the constructions of the quasi regular prisms and their duals in 3D dimensions with the use of 3D Coxeter diagrams.