Quasi-equivalence is part of a wider set of structural constraints on virus architecture. (a) A large number of viruses exhibit icosahedral symmetry in the organization of their protein containers, i.e. they share a set of five-, three- and twofold axes with an icosahedron. Here, four example symmetry axes are shown on a viral protein container. (b) The T = 4 surface lattice from quasi-equivalence theory that encodes the protein organizations in capsids composed of 240 coat proteins. (c), (d) A packing of overlapping icosahedra generates a partition of the icosahedral face akin to the T = 4 structure: (c) shows three of the 60 translated and rotated icosahedra as solids; (d) shows the edges of these icosahedra, together with the vertices of all 60 icosahedra. This is an example of the structural constraints implied by our theory.