Explicit construction of the Voronoi and Delaunay cells ofW(A_n) andW(D_n) lattices and their facets

Abstract

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.

Quantum gravity at the fifth root of unity

“…This representation theory is as well understood as that of SU (2). There are 3 independent Casimir invariants for each face 5 and Y a = (L − a ) 5 . When X a and Y a have zero eigenvalues for all vectors in the linear vector space carrying the irreducible representation, the representations have classical analogs, and can be constructed by specializing from the general values of q to the specific root of unity.…”

“…The tetrahedron H j q1 × H j q2 × H j q3 × H j q4 is 5 dimensional. The dimension grows with increasing number of representations, following the well known Fibonacci sequence, which gives the name of Fibonacci anyons for the representations of the SU (2) 5 q symmetry in context of topological phases of matter and quantum computing field of researches. In this language the A v = W 5 1 (j lv ) building block amplitude is just the known F symbol that appears in those anyonic models [15,16].…”

“…For a review of the motivations and main results of LQG and spin foam models we recommend the book [1]. We also recommend a review of the main elements of the higher dimensional Lie algebra unification program [4,5,6,7,8], which is closely related to string theory. With those results as guides, this paper will construct a model that does not have a priori assumptions of general relativity (GR) and the usual Yang Mills gauge quantum fields, but yet has their building blocks arXiv:1903.10851v2 [hep-th] 15 Apr 2019 built-in in a consistent way to generate their emergent macroscopic properties starting only from the notion of quantum geometry.…”

Icosahedral Polyhedra from D6 Lattice and Danzer’s ABCK Tiling

“…The Voronoi cell of the root lattice D 6 is the dual polytope of the root polytope of ω 2 [15] and determined as the union of the orbits of the weights ω 1 , ω 5 and ω 6 , which correspond to the holes of the root lattice [16]. If the roots and weights of H 3 are defined in two complementary 3D spaces E and E ⊥ as β i (β i ) and v i (v i ), (i = 1, 2, 3), respectively, then they can be expressed in terms of the roots and weights of D 6 as…”

Dodecahedral structures with Mosseri–Sadoc tiles

“…Since the Delone cells tile the root lattice in an alternating order (Conway & Sloane, 1999) it is expected that the tiles projected from the Delone cells may tile the 3D Euclidean space in an aperiodic manner with an icosahedral symmetry. See the work of Koca et al (2018) for a detailed exposition of the root lattice D 6 with a discussion of the Voronoi cell and the Delone polytopes. The dual of the root polytope of D 6 determined as the orbit of the weight vector !…”