Icosahedral symmetry breaking: C 60 to C 78 , C 96 and to related nanotubes

Exact icosahedral symmetry of C 60 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 A 2 because it is isomorphic to the Weyl group of the simple Lie algebra A 2 . Eight of the A 2 orbits are hexagons and four are triangles. Only two of the hexagons appear as part of the C 60 surface shell. The orbits form a stack of parallel layers centered on the axis of C 60 passing through the centers of two opposite hexagons on the surface of C 60 . By inserting into the middle of the stack two A 2 orbits of six points each and two A 2 orbits of three points each, one can match the structure of C 78 . Repeating the insertion, one gets C 96 ; 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.