**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 Bœhm, 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 Bœhm, 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).