Abstract: We explore the possibility that extended icosahedral systems occurring in nature may be describable by an extension of the icosahedral group by a non-compact operation. We derive such extensions in two ways, firstly by direct Kac-Moody-type extension of the (non-crystallographic) H3 root system [1], and secondly by projection of the affine (crystallographic) D6 root system [2]. We discuss applications to the structure of viruses as well as that of nested fullerene shells, so called carbon onions [3]. Time permitting, I will briefly review how rank 3 root systems induce the exceptional rank 4 root systems via a novel Clifford spinor construction [4].

[1] Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups. P-P Dechant, C Bhm, R Twarock: J. Phys. A: Math. Theor. 45, 285202 (2012).
[2] Affine extensions of non-crystallographic Coxeter groups induced by projection. P-P Dechant, C Bhm, R Twarock: Journal of Mathematical Physics 54, 093508 (2013).
[3] Viruses and Fullerenes Symmetry as a Common Thread? P-P Dechant, J Wardman, T Keef, R Twarock: Acta Crystallographica A 70, 162-167 (2014).
[4] Platonic Solids generate their 4-dimensional analogues, P-P Dechant: Acta Crystallographica A 69 (6). pp. 592-602. (2013).