Catalan Solids are the duals of the Archimedean solids, vertices of which can be obtained from the Coxeter-Dynkin diagrams 3 () WH acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result as the orbits derived from fundamental weights. The Platonic solids are dual to each others however duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), (001) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011) and (111)

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

Vertices of the 4-dimensional semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the E 8 root system which decomposes into two copies of the root system of H 4 . The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group W (H 4 ). It is the group Aut(H 2 ⊕H ′ 2 ) which is constructed in terms of 20 quaternionic roots of the Coxeter diagram H 2 ⊕ H ′ 2 . The root system of H 4 represented by the binary icosahedral group I of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram H 2 ⊕ H ′ 2 are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the grand antiprism. We give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions. The dual polytope of the grand antiprism has been also constructed.

Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter-Weyl groups W(A) and W(D) are constructed. The face-centred cubic (f.c.c.) and body-centred cubic (b.c.c.) lattices are obtained in this context. Basic definitions are introduced such as parallelotope, fundamental simplex, contact polytope, root polytope, Voronoi cell, Delone cell, n-simplex, n-octahedron (cross polytope), n-cube and n-hemicube and their volumes are calculated. The Voronoi cell of the root lattice is constructed as the dual of the root polytope which turns out to be the union of Delone cells. It is shown that the Delone cells centred at the origin of the root lattice A are the polytopes of the fundamental weights ω, ω,…, ω and the Delone cells of the root lattice D are the polytopes obtained from the weights ω, ω and ω. A simple mechanism explains the tessellation of the root lattice by Delone cells. It is proved that the (n-1)-facet of the Voronoi cell of the root lattice A is an (n-1)-dimensional rhombohedron and similarly the (n-1)-facet of the Voronoi cell of the root lattice D is a dipyramid with a base of an (n-2)-cube. The volume of the Voronoi cell is calculated via its (n-1)-facet which in turn can be obtained from the fundamental simplex. Tessellations of the root lattice with the Voronoi and Delone cells are explained by giving examples from lower dimensions. Similar considerations are also worked out for the weight lattices A* and D*. It is pointed out that the projection of the higher-dimensional root and weight lattices on the Coxeter plane leads to the h-fold aperiodic tiling, where h is the Coxeter number of the Coxeter-Weyl group. Tiles of the Coxeter plane can be obtained by projection of the two-dimensional faces of the Voronoi or Delone cells. Examples are given such as the Penrose-like fivefold symmetric tessellation by the A root lattice and the eightfold symmetric tessellation by the D root lattice.

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. a)