Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)

TL;DR

This paper constructs the snub 24-cell, a unique 4D chiral polytope, using quaternionic representations of Coxeter-Weyl groups, and explores its symmetry, dual, and relation to other Coxeter groups.

Contribution

It introduces a quaternionic approach to derive the snub 24-cell and its symmetry group, revealing new geometric and algebraic properties of this 4D polytope.

Findings

Vertices obtained as orbits of the Coxeter-Weyl group

Mirror images combine into a quasi-regular 4D polytope

Dual of the snub 24-cell constructed and related to other Coxeter groups

Abstract

Snub 24-cell is the unique uniform chiral polytope in four dimensions consisting of 24 icosahedral and 120 tetrahedral cells. The vertices of the 4-dimensional semi-regular polytope snub 24-cell and its symmetry group ${(W(D_{4})\mathord{/{\vphantom {(W(D_{4}) C_{2}}}. \kern-\nulldelimiterspace} C_{2}}):S_{3} $ of order 576 are obtained from the quaternionic representation of the Coxeter-Weyl group \textbf{$W(D_{4}).$}The symmetry group is an extension of the proper subgroup of the Coxeter-Weyl group \textbf{$W(D_{4})$}by the permutation symmetry of the Coxeter-Dynkin diagram \textbf{$D_{4} .$} The 96 vertices of the snub 24-cell are obtained as the orbit of the group when it acts on the vector \textbf{$\Lambda =(\tau, 1, \tau, \tau)$}or\textbf{}on the vector\textbf{$\Lambda =(\sigma, 1, \sigma, \sigma)$}in the Dynkin basis with\textbf{$\tau =\frac{1+\sqrt{5}}{2} {\rm and}\sigma =\frac{1-\sqrt{5}}{2} {\rm .}$} The two different sets represent the mirror images of the snub 24-cell. When two mirror images are combined it leads to a quasi regular 4D polytope invariant under the Coxeter-Weyl group \textbf{$W(F_{4}).$}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 \textbf{$W(H_{4}){\rm and}W(E_{8})$}has been pointed out.