  We clarify the relationships between the eight structural types of toroidal
carbon nanotubes (TCNTs), which can be identified as the eight corners of a
cube of structural transformation. The four families with Dnh symmetry can be
related by rim rotations, and the same is true for those with Dnd symmetries.
These two sets are then connected by horizontal shiftings, thereby completing
the cube. Moreover, we further point out that there are five more highly
symmetric Dnh structural types that can be derived from performing the
generalized Stone-Wales transformatoin on certain TCNTs with Dnh structural
types.
