Quaternionic representation of snub 24-cell and its dual polytope derived from [formula omitted] root system

Abstract

Vertices of the 4-dimensional semi-regular polytope, snub 24-cell and its symmetry group ( W ( D 4 ) / C 2 ) : S 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 proper rotation subgroup of the Weyl group of D 4 with symmetric group of order 3 denoted by ( W ( D 4 ) / C 2 ) : S 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 2 ) : S 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 2 ) : S 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 2 ) : S 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 2 ) : S 3 are given in the appendix.