Towards a classification of icosahedral viruses in terms of indexed polyhedra

The standard Caspar & Klug classification of icosahedral viruses by means of triangulation numbers and the more recent novel characterization of Twarock leading to a Penrose-like tessellation of the capsid of viruses not obeying the Caspar-Klug rules can be obtained as a special case in a new approach to the morphology of icosahedral viruses. Considered are polyhedra with icosahedral symmetry and rational indices. The law of rational indices, fundamental for crystals, implies vertices at points of a lattice (here icosahedral). In the present approach, in addition to the rotations of the icosahedral group 235, crystallographic scalings play an important rôle. Crystallographic means that the scalings leave the icosahedral lattice invariant or transform it to a sublattice (or to a superlattice). The combination of the rotations with these scalings (linear, planar and radial) permits edge, face and vertex decoration of the polyhedra. In the last case, satellite polyhedra are attached to the vertices of a central polyhedron, the whole being generated by the icosahedral group from a finite set of points with integer indices. Three viruses with a polyhedral enclosing form given by an icosahedron, a dodecahedron and a triacontahedron, respectively, are presented as illustration. Their cores share the same polyhedron as the capsid, both being in a crystallographic scaling relation.