This fullerene contains nine pairs of symmetry-equivalent six-membered rings preferred for the beginning of the numbering according to rule Fu-3.4.1. They are related by the center of inversion (shown by a bold dot in Fig. 31a). Symmetry-equivalent rings are identified by the same letter, unprimed and primed (in Fig. 31a primed letters indicate rings away from the viewer and bold letters rings closer to the viewer). Because of symmetry only one set of nine nonequivalent six-membered rings is considered (those denoted by unprimed letters). Each of these rings is a priori eligible to begin the numbering, and they are ranked on the basis of their atoms (Fu-3.4.1). Rings B, C, and D contain four contiguous 6,6,6 atoms and two 6,6,5 atoms, and are preferred for beginning the numbering. Please note that some of the other rings also contain four 6,6,6 and two 6,6,5 atoms, but the 6,6,6 atoms are not contiguous. Since the pathways having four contiguous 6,6,6 atoms at position 1-4 would be preferred according to Fu-3.4.2, they are examined first for contiguous numbering to reduce the number of necessary inspections. The 6,6,6 atoms are indicated by the letters a to g in Fig. 31b. There are six pathways to be considered: a to b to c to d and d to c to b to a, from ring C; b to c to f to g and g to f to c to b, from ring B; f to c to d to e and e to d to c to f, from ring D. Only one of these pathways (d to c to b to a, from ring C) leads to a contiguous numbering, that is shown in Fig. 31c.
Fig. 31. Systematic numbering of (C56-Ci)[5,6]fullerene
5. P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995