Grand antiprism and quaternions

Vertices of the four-dimensional (4D) semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the E ₈ root system which decomposes into two copies of the root system of H ₄. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group W(H ₄). It is the group Aut(H ₂ ⊕ H′ ₂) which is constructed in terms of 20 quaternionic roots of the Coxeter diagram H ₂ ⊕ H′ ₂. The root system of H ₄ represented by the binary icosahedral group I of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram H ₂ ⊕ H′ ₂ are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the grand antiprism. We give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions. The dual polytope of the grand antiprism has also been constructed.