Icosahedral symmetry breaking: C₆₀ to C₇₈, C₉₆ and to related nanotubes

Exact icosahedral symmetry of C₆₀ is viewed as the union of 12 orbits of the symmetric subgroup of order 6 of the icosahedral group of order 120. Here, this subgroup is denoted by A₂ because it is isomorphic to the Weyl group of the simple Lie algebra A₂. Eight of the A₂ orbits are hexagons and four are triangles. Only two of the hexagons appear as part of the C₆₀ surface shell. The orbits form a stack of parallel layers centered on the axis of C₆₀ passing through the centers of two opposite hexagons on the surface of C₆₀. By inserting into the middle of the stack two A₂ orbits of six points each and two A₂ orbits of three points each, one can match the structure of C₇₈. Repeating the insertion, one gets C₉₆; multiple such insertions generate nanotubes of any desired length. Five different polytopes with 78 carbon-like vertices are described; only two of them can be augmented to nanotubes.