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In two series of papers we construct quasi regular polyhedra and their dualswhich are similar to the Catalan solids. The group elements as well as thevertices of the polyhedra are represented in terms of quaternions. In thepresent paper we discuss the quasi regular polygons (isogonal and isotoxalpolygons) using 2D Coxeter diagrams. In particular, we discuss the isogonalhexagons, octagons and decagons derived from 2D Coxeter diagrams and obtainaperiodic tilings of the plane with the isogonal polygons along with theregular polygons. We point out that one type of aperiodic tiling of the planewith regular and isogonal hexagons may represent a state of graphene where onecarbon atom is bound to three neighboring carbons with two single bonds and onedouble bond. We also show how the plane can be tiled with two tiles; one ofthem is the isotoxal polygon, dual of the isogonal polygon. A general method isemployed for the constructions of the quasi regular prisms and their duals in3D dimensions with the use of 3D Coxeter diagrams.

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