  Icosahedral virus capsids are composed of symmetrons, organized arrangements
of capsomers. There are three types of symmetrons: disymmetrons, trisymmetrons,
and pentasymmetrons, which have different shapes and are centered on the
icosahedral 2-fold, 3-fold and 5-fold axes of symmetry, respectively. In 2010
[Sinkovits & Baker] gave a classification of all possible ways of building an
icosahedral structure solely from trisymmetrons and pentasymmetrons, which
requires the triangulation number T to be odd. In the present paper we
incorporate disymmetrons to obtain a geometric classification of icosahedral
viruses formed by regular penta-, tri-, and disymmetrons. For every class of
solutions, we further provide formulas for symmetron sizes and parity
restrictions on h, k, and T numbers. We also present several methods in which
invariants may be used to classify a given configuration.
