4D Pyritohedral Symmetry with Quaternions, Related Polytopes and Lattices

TL;DR

This paper extends pyritohedral symmetry to four dimensions using quaternions, explores related polytopes and lattices, and discusses implications for crystallography and quasicrystals.

Contribution

It introduces a quaternionic representation of 4D pyritohedral symmetry and connects it to known 4D polytopes and lattices, expanding symmetry understanding in higher dimensions.

Findings

Extended pyritohedral symmetry to 4D using quaternions.

Identified related 4D polytopes including the pseudo snub 24-cell.

Connected the symmetry group to the root lattice of W(D4).

Abstract

We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present the group elements in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W(F4) and W(H4) implying that it is a group relevant to the crystallography as well as quasicrystallographic structures in 4-dimensions. First we review the pyritohedral symmetry in 3 dimensional Euclidean space which is a maximal subgroup both in the Coxeter-Weyl groups W(B3)=Aut(D3) and W(H3). The related polyhedra in 3-dimensions are the two dual polyhedra pseudoicosahedron- pyritohedron and the pseudo icosidodecahedron. In quaternionic representations it finds a natural extension to the 4-dimensions.The related polytopes turn out to be the pseudo snub 24-cell and its dual polytope expressed in terms of a parameter x leading to snub 24-cell and its dual in the limit where the parameter x takes the golden ratio. It turns out that the relevant lattice is the root lattice of W(D4).