Snub 24-cell derived from the Coxeter-Weyl group W(D ₄)

Snub 24-cell is the unique uniform chiral polytope in four dimensions consisting of 24 icosahedral and 120 tetrahedral cells. The vertices of the four-dimensional semiregular polytope snub 24-cell and its symmetry group (W(D ₄)/C ₂): S ₃ of order 576 are obtained from the quaternionic representation of the CoxeterWeyl group W(D ₄). The symmetry group is an extension of the proper subgroup of the CoxeterWeyl group W(D ₄) by the permutation symmetry of the CoxeterDynkin diagram D ₄. The 96 vertices of the snub 24-cell are obtained as the orbit of the group when it acts on the vector Λ = (τ, 1, τ, τ) or on the vector Λ = (σ, 1, σ, σ) in the Dynkin basis with τ = 1+√5/2 and σ = 1-√5/2. The two different sets represent the mirror images of the snub 24-cell. When two mirror images are combined it leads to a quasiregular four-dimensional polytope invariant under the CoxeterWeyl group W(F ₄). Each vertex of the new polytope is shared by one cube and three truncated octahedra. Dual of the snub 24-cell is also constructed. Relevance of these structures to the Coxeter groups W(H ₄) and W(E ₈) has been pointed out.