Explicit construction of the voronoi and delaunay cells of W(A_n) and W(D_n) lattices and their facets

Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter-Weyl groups W(A_n) and W(D_n) 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_n are the polytopes of the fundamental weights ω₁, ω₂, …, ω_n and the Delone cells of the root lattice D_n are the polytopes obtained from the weights ω₁, ω_n-1 and ω_n. 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_n is an (n-1)- dimensional rhombohedron and similarly the (n-1)-facet of the Voronoi cell of the root lattice D_n 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_n* and D_n*. 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 A₄ root lattice and the eightfold symmetric tessellation by the D₅ root lattice.