Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups

TL;DR

This paper introduces new affine extensions of non-crystallographic Coxeter groups using a Kac-Moody-type approach, revealing their role in generating symmetries relevant to quasicrystals, viruses, and fullerenes.

Contribution

It presents novel asymmetric affine extensions of H_2, H_3, and H_4 Coxeter groups, classifies associated translations, and connects these to physical applications.

Findings

Affine reflection planes generate translations along symmetry axes.

Translations classified via Fibonacci recursion.

Provides a group-theoretic explanation for icosahedral symmetry extensions.

Abstract

Motivated by recent results in mathematical virology, we present novel asymmetric Z[tau]-integer-valued affine extensions of the non-crystallographic Coxeter groups H_2, H_3 and H_4 derived in a Kac-Moody-type formalism. In particular, we show that the affine reflection planes which extend the Coxeter group H_3 generate (twist) translations along 2-, 3- and 5-fold axes of icosahedral symmetry, and we classify these translations in terms of the Fibonacci recursion relation applied to different start values. We thus provide an explanation of previous results concerning affine extensions of icosahedral symmetry in a Coxeter group context, and extend this analysis to the case of the non-crystallographic Coxeter groups H_2 and H_4. These results will enable new applications of group theory in physics (quasicrystals), biology (viruses) and chemistry (fullerenes).