The vast majority of viruses infecting humans, animals and plants
package their genomic material into protein containers that are organised
according to icosahedral symmetry. The structure, formation and strategies
of infection of these viruses can therefore be understood in terms of group
theoretical tools. We have introduced and classified affine extensions of
non-crystallographic Coxeter groups to predict details of virus
architecture, including the dimensions and characteristic structural
features of the protein containers and the organisation of the genomic
material within. In this talk, these new mathematical tools will be
introduced and their predictive power for virus architecture demonstrated
based on a number of test viruses. It will then be demonstrated that these
mathematical insights shed new light on how viruses form and infect their
hosts.