We show that the full three-dimensional architecture of simple viruses is encoded by affine symmetries. In particular, the locations and structures of all material boundaries can be predicted with our method, including the organisation of the capsid proteins and the genomic material. This approach departs radically from the 2-dimensional schematic representations of viral capsids in Caspar-Klug Theory and its recent generalisations, and opens up new possibilities to study the immuno-dominant epitopes and viral evolution. Applications to the modelling of virus assembly are also discussed, and we show that the counting of assembly pathways can be cast into a Hamiltonian paths problem