This fullerene features one C3 axis connecting the center of a six-membered ring at the near end and an atom at the intersection of three six-membered rings at the far end (Fig. 22a). For symmetry reasons two pathways starting in the ring must be considered: a to a' to a" and a' to a" to a'''. They both lead to numberings that become discontiguous at position 81, as shown in Figs. 22b and 22c, respectively. The pathway starting from the atom z at the far end of the axis leads to a numbering discontiguous at position 66 (not shown), that is discarded on the basis of Fu-3.2.1. Selection between the remaining two numberings is made on the basis of Fu-3.2.3. By comparing the ranking of the atoms of the two sequences (cf. Fu-3.1.4), it is found that the atom at position 8 of the numbering in Fig. 22b is a 6,6,6 atom whereas the numbering shown in Fig. 22c features a 6,6,5 atom at the same position. The former is therefore preferred. The completion of the numbering is obvious, because the unassigned site is reduced to a final isolated unnumbered atom which receives the last number in the sequence.
Fig. 22. Systematic numbering of (C82-C3v)[5,6]fullerene
5. P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995