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)/C2):S3 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 proper rotation subgroup of the Weyl group of D4 with symmetric group of order 3 denoted by (W(D4)/C2):S3, 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)/C2):S3. The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group (W(D4)/C2):S3. 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)/C2):S3. 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)/C2):S3 are given in the appendix.