Quaternionic Representation of Snub 24-Cell and its Dual Polytope Derived From E8 Root System

Vertices of the 4-dimensional semi-regular polytope, snub 24-cell and its symmetry group W (D4) : C3 of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of E8 root system. A simple method is employed to construct the E8 root system in terms of icosians which decomposes into two copies of the quaternionic root system of the Coxeter group W (H4), 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 H4 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 (H4) as well as W (F4). It is noted that the group is isomorphic to the semi-direct product of the Weyl group of D4 with the cyclic group of order 3 denoted by W (D4) : C3, 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 (H4) under the orbits of W (D4) : C3. The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group W (D4) : C3. 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 (D4) : C3. The dual polytope of the snub 24-cell is explicitly constructed. Decompositions of the Archimedean W (H4) polytopes under the symmetry of the group W (D4) : C3 are given in the appendix.