Mitsuho Yoshida and Patrick W. Fowler
A general geometrical construction for fullerenes of threefold dihedral (D3, D3h, D3d) symmetry is reported and used to find those examples of this symmetry that are without face spirals. The smallest, a D3 384-atom cage, has its twelve pentagons arranged in three separate crosses of four fully fused rings and has four atoms more than the smallest known non-spirallable fullerene. Another D3 structure, a 672-atom cage, is the smallest example of an isolated-pentagon fullerene without a spiral, improving on the previous upper bound by 128 atoms. Altogether, 61 new enantiomeric pairs of nonspirallable fullerenes with less than 1000 atoms belonging to four different classes, all of D3 symmetry, are identified.