3.2.7. Systematic numbering for (C₈₀-I_h)[5,6]fullerene

This fullerene features six symmetry-equivalent C₅ axes (connecting midpoints of opposite five-membered rings) ten symmetry-equivalent C₃ axes (connecting atoms at the intersection of three six-membered rings), and fifteen symmetry-equivalent C₂ axes (passing through the midpoints of opposite hexagons). These axes are shown in Fig. 21a. When one of the C₅ axes is used as the reference axis, the pathway can begin at any one of the five symmetry-equivalent atoms of the pentagon (indicated as a in Fig. 21a); the numbering (shown in Fig. 21b) becomes discontiguous at position 75. When one of the C₃ axes is used as the reference axis only one pathway from atom b must be considered; this pathway becomes discontiguous at position 66, and is therefore discarded (Fu-3.2.3). When one of the C₂ axes is used as the reference axis there are three pathways to be considered: c to d to e; d to e to c'; e to c' to d'. One of these leads to the numbering becoming discontiguous at position 75 (not shown); the others lead to numberings in which the discontiguities occur at positions 72 and 66. According to Fu-3.2.1 the three numberings built around the C₂ axis are discarded. The correct numbering is therefore that shown in Fig. 21b. The completion of the numbering by application of Fu-3.2.4 is straightforward.

Fig. 21. Systematic numbering of (C₈₀-I_h)[5,6]fullerene

5. P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995