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

Abstract

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(H4). The maximal subgroups W(SU(5)):Z2 and W(H3) × Z2 of W(H4) 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(H3) 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.

Regular and Irregular Chiral Polyhedra from Coxeter Diagrams via Quaternions

“…Substituting the first solution for = τ which results in α = 0, β = τ, γ = τ 2 , the set of vectors in (28) can be written as: (29) are transformed to each other. This proves that the pyritohedral group T, ±T can be extended to the icosahedral group I, ±I by the generators so that the set of vertices in (29) is invariant under the icosahedral group of order 120.…”

“…It is well known that mid-points of the edges of a regular icosahedron or dodecahedron form the Archimedean solid icosidodecahedron with 30 vertices 32 faces (12 pentagons + 20 triangles) and 60 edges which can be obtained from the Coxeter graph of as an orbit (010) [28]. An irregular icosidodecahedron consists of irregular pentagonal faces and scalene triangles in the most general case and will be derived from the chiral tetrahedral group and will be extended by pyritohedral group representing a larger symmetry.…”

“…Imposing the pyritohedral group invariance (x = y) and moreover, letting x = τ and dividing each vector by a scale factor 2τ 2 , one obtains the usual quaternionic vertices of the icosidodecahedron [ Here, with , with and define three orbits under the chiral tetrahedral symmetry of sizes 12, 12 and 6, respectively. Imposing the pyritohedral group invariance ( = ) and moreover, letting = and dividing each vector by a scale factor 2 , one obtains the usual quaternionic vertices of the icosidodecahedron [28] 1…”

SNUB 24-CELL DERIVED FROM THE COXETER–WEYL GROUP W(D4)

“…Let us recall that the set of 120 unit quaternions I = T ⊕ S represents the elements of the binary icosahedral group as well as the vertices of the Platonic polytope 600-cell [14] consisting of 600 tetrahedra, the symmetry of which is the [17,18], this analogy can be traced back to the Coxeter-Weyl group W (E 8 ⊕ E 8 ) and perhaps to the heterotic superstring theory. Now we come back to the structure of the snub 24-cell.…”

Catalan solids derived from three-dimensional-root systems and quaternions

“…represents the Coxeter group of order 14,400, the symmetry of the 120-cell and the 600-cell [13]. When qp is substituted in (4) (12) (even number of (-) sign) where Let us take one of the vertex of the truncated tetrahedron, say, the highest weight…”