Grand Antiprism and Quaternions

TL;DR

This paper represents the vertices and symmetry group of the grand antiprism, a 4D semi-regular polytope, using quaternions, and analyzes its structure and dual in relation to the 600-cell and Coxeter groups.

Contribution

It introduces a quaternionic representation of the grand antiprism's vertices and symmetry, linking it to the E8 root system and Coxeter groups, and details its cell construction and dual.

Findings

Vertices of the grand antiprism are represented using quaternions.

The symmetry group is identified as a maximal subgroup of W(H4).

The dual polytope of the grand antiprism is constructed.

Abstract

Vertices of the 4-dimensional semi-regular polytope, the \textit{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 \textbf{$E_{8} $} root system which decomposes into two copies of the root system of $H_{4} $. The symmetry of the \textit{grand antiprism} is a maximal subgroup of the Coxeter group $W(H_{4})$. It is the group $Aut(H_{2} \oplus H'_{2})$ which is constructed in terms of 20 quaternionic roots of the Coxeter diagram $H_{2} \oplus H'_{2}$. The root system of $H_{4} $ represented by the binary icosahedral group \textit{I}of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram $H_{2} \oplus H'_{2}$ are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the\textit{grand antiprism}. We give a detailed analysis of the construction of the cells of the\textit{grand antiprism} in terms of quaternions. The dual polytope of the \textit{grand antiprism} has been also constructed.