  In a seminal paper Caspar and Klug established a theory that provides a
family of polyhedra as blueprints for the structural organisation of viral
capsids. In particular, they encode the locations of the proteins in the shells
that encapsulate, and hence provide protection for, the viral genome. Despite
of its huge success and numerous applications in virology experimental results
have provided evidence for the fact that the theory is too restrictive to
describe all known viruses. Especially, the family of Papovaviridae, which
contains cancer-causing viruses, falls out of the scope of this theory.
  In a recent paper we have shown that certain members of the family of
Papovaviridae can be described via tilings. In this paper, we develop a
comprehensive mathematical framework for the derivation of all surface
structures of viral particles in this family. We show that this formalism fixes
the structure and relative sizes of all particles collectively so that there
exists only one scaling factor that relates the sizes of all particles with
their biological counterparts.
  The series of polyhedra derived here complements the Caspar-Klug family of
polyhedra. It is the first mathematical result that provides a common
organisational principle for different types of viral particles in the family
of Papovaviridae and paves the way for an understanding of Papovaviridae
polymorphism. Moreover, it provides crucial input for the construction of
assembly models.
