4d-polytopes described by Coxeter diagrams and quaternions (original) (raw)

Group theoretical analysis of 600-cell and 120-cell 4D polytopes with quaternions

Journal of Physics A: Mathematical and Theoretical, 2007

600-cell {3, 3, 5} and 120-cell {5, 3, 3} four-dimensional dual polytopes relevant to quasicrystallography have been studied with the quaternionic representation of the Coxeter group W (H 4 ). The maximal subgroups W (SU(5)) : Z 2 and W (H 3 ) × Z 2 of W (H 4 ) play important roles in the analysis of cell structures of the dual polytopes. In particular, the Weyl-Coxeter group W (SU(4)) is used to determine the tetrahedral cells of the polytope {3, 3, 5}, and the Coxeter group W (H 3 ) is used for the dodecahedral cells of {5, 3, 3}. Using the Lie algebraic techniques in terms of quaternions, we explicitly construct cell structures forming the vertices of the 4D polytopes. PACS numbers: 61.44.Br, 02.20.Sv

Self-Dual Regular 4-Polytopes and Their Petrie-Coxeter-Polyhedra

Results in Mathematics, 1987

Coxeter's regular skew polyhedra in euclidean 4-space IE4 are intimately related to the self-dual regular 4-polytopes. The same holds for two of the three Petrie-Coxeter-polyhedra and the (self-dual) cubical tessellation in IE 3. In this paper we discuss the combinatorial Petrie-Coxeter-polyhedra associated with the self-dual regular 4-incidence-polytopes.

Grand antiprism and quaternions

Journal of Physics A-mathematical and General, 2009

Vertices of the four-dimensional (4D) 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 E8 root system which decomposes into two copies of the root system of H4. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group W(H4). It is the group Aut(H2 ⊕ H'2) which is constructed in terms of 20 quaternionic roots of the Coxeter diagram H2 ⊕ H'2. The root system of H4 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 H2 ⊕ 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 also been constructed.

Quaternionic Representations of the Pyritohedral Group, Related Polyhedra and Lattices

arXiv: Mathematical Physics, 2015

We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) lattices as the root and the weight lattices of the affine Coxeter groups W(D3) and W(B3)=Aut(D3). The rank-3 Coxeter-Weyl groups describing the point tetrahedral symmetry and the octahedral symmetry of the cubic lattices have been constructed in terms of quaternions. Reflection planes of the Coxeter-Dynkin diagrams are identified with certain planes of the unit cube. It turns out that the pyritohedral symmetry takes a simpler form in terms of quaternionic representation. The D3 diagram is used to construct the vertices of polyhedra relevant to the cubic lattices and, in particular, constructions of the pseudoicosahedron and its dual pyritohedron are explicitly worked out.

Quaternionic representation of snub 24-cell and its dual polytope derived from root system

Linear Algebra and its Applications, 2011

Vertices of the 4-dimensional semi-regular polytope, snub 24-cell and its symmetry group W (D 4 ) : C 3 of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of E 8 root system. A simple method is employed to construct the E 8 root system in terms of icosians which decomposes into two copies of the quaternionic root system of the Coxeter group W (H 4 ), while one set is the elements of the binary icosahedral group the other set is a scaled copy of the first. The quaternionic root system of H 4 splits as the vertices of 24-cell and the snub 24-cell under the symmetry group of the snub 24-cell which is one of the maximal subgroups of the group W (H 4 ) as well as W (F 4 ). It is noted that the group is isomorphic to the semi-direct product of the Weyl group of D 4 with the cyclic group of order 3 denoted by W (D 4 ) : C 3 , the Coxeter notation for which is [3, 4, 3 + ]. We analyze the vertex structure of the snub 24-cell and decompose the orbits of W (H 4 ) under the orbits of W (D 4 ) : C 3 . The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group W (D 4 ) : C 3 . In particular, it has been shown that the dual polytopes 600-cell with 120 vertices and 120-cell with 600 vertices decompose as 120=24+96 and 600=24+96+192+288 respectively under the group W (D 4 ) : C 3 . The dual polytope of the snub 24-cell is explicitly constructed. Decompositions of the Archimedean W (H 4 ) polytopes under the symmetry of the group W (D 4 ) : C 3 are given in the appendix.

Four-Dimensional Regular Polyhedra

Discrete & Computational Geometry, 2007

This paper completes the classification of the four-dimensional (finite) regular polyhedra, of which those with planar faces were-in effect-found by Arocha, Bracho and Montejano. However, the methods employed here are in the same spirit as those used in the description of all three-dimensional regular polytopes by this author and Schulte, and the regular polytopes of full rank by this author. The procedure has two stages. First, the possible dimension vectors (dim R 0 , dim R 1 , dim R 2) of the mirrors R 0 , R 1 , R 2 of the generating reflexions of the symmetry groups are determined. Second, all polyhedra with a given dimension vector are found. Most of the polyhedra are related to four-dimensional Coxeter groups, although one class has to be approached using quaternions.