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.

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell [Formula: see text] and the rectified 5-cell [Formula: see text] derived from the [Formula: see text] Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope [Formula: see text] whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the [Formula: see text] charge conservation. The Dynkin diagram symmetry of the [Formula: see text] diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes [Formula: see text] whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like tiling of the plane which can be used for the description of the 5-fold symmetric quasicrystallography. The model can be extended to [Formula: see text] and even to [Formula: see text] by noting the Coxeter–Dynkin diagram embedding [Formula: see text]. Another embedding can be made through the relation [Formula: see text] for more popular [Formula: see text]. Appendix A includes the quaternionic representations of the Coxeter–Weyl groups [Formula: see text] which can be obtained directly from [Formula: see text] by projection. This leads to relations of the [Formula: see text] polytopes with the quasicrystallography in 4D and [Formula: see text] polytopes. Appendix B discusses the branching of the polytopes in terms of the irreducible representations of the Coxeter–Weyl group [Formula: see text].

We exploit the fact that two-dimensional facets of the Voronoi and Delone cells of the root lattice are identical rhombuses and equilateral triangles respectively. Their orthogonal projections onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman-Beenker tiles, Robinson triangles and Danzer triangles to name a few. We point out that the dihedral subgroup of order 2ℎ involving the Coxeter element of order ℎ = + 1 of the Coxeter-Weyl group plays a crucial role for ℎ-fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4, 5, 6, 7, 8 and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice 3 whose Wigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an ℎ = 4 fold symmetry.

It is well known that the point group of the root lattice D6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H3, its roots, and weights are determined in terms of those of D6. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D6 determined by a pair of integers (m1, m2) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H3, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m1, m2) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H3, and those related transformations in 6D space with D6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D6. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed.

The 3D facets of the Delone cells of the root lattice D 6 which tile the 6D Euclidean space in an alternating order are projected into 3D space. They are classified into six Mosseri–Sadoc tetrahedral tiles of edge lengths 1 and golden ratio τ = (1 + 51/2)/2 with faces normal to the fivefold and threefold axes. The icosahedron, dodecahedron and icosidodecahedron whose vertices are obtained from the fundamental weights of the icosahedral group are dissected in terms of six tetrahedra. A set of four tiles are composed from six fundamental tiles, the faces of which are normal to the fivefold axes of the icosahedral group. It is shown that the 3D Euclidean space can be tiled face-to-face with maximal face coverage by the composite tiles with an inflation factor τ generated by an inflation matrix. It is noted that dodecahedra with edge lengths of 1 and τ naturally occur already in the second and third order of the inflations. The 3D patches displaying fivefold, threefold and twofold symmetries are obtained in the inflated dodecahedral structures with edge lengths τ n with n ≥ 3. The planar tiling of the faces of the composite tiles follows the edge-to-edge matching of the Robinson triangles.