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

Abstract

Motivated by recent results in mathematical virology, we present novel asymmetric -integer-valued affine extensions of the non-crystallographic Coxeter groups H2, H3 and H4 derived in a Kac–Moody-type formalism. In particular, we show that the affine reflection planes which extend the Coxeter group H3 generate (twist) translations along two-, three- and five-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 H2 and H4. These results will enable new applications of group theory in physics (quasicrystals), biology (viruses) and chemistry (fullerenes).