3.2.11. Systematic numbering for (C₁₂₀-I_h)[4,6,10]fullerene

This fullerene features six symmetry-equivalent C₅ axes, ten symmetry-equivalent C₃ axes, and fifteen symmetry-equivalent C₂ axes. The C₅ axes connect midpoints of opposite ten-membered rings; the C₃ axes connect midpoints of opposite six-membered rings; the C₂ axes connect midpoints of opposite four-membered rings (Fig. 25a). Even if all the atoms in this fullerene are symmetry-equivalent, different lettering is maintained for clarity. For symmetry reasons there are two pathways to be considered in the ten-membered ring: a to a" and a' to a. Of these, the a to a" one leads to a numbering that becomes discontiguous at position 60 (Fig. 25b); the a' to a one leads to a numbering discontiguity at position 30. For symmetry reasons, there are two pathways to be considered in the six-membered ring: b to b' and b' to b". Of these, the b to b' pathway leads to a numbering that becomes discontiguous at position 30; the b' to b" pathway leads to a numbering discontiguous at position 60 (Fig. 25c). For symmetry reasons, there are two pathways to be considered in the four-membered ring: a' to a and a to c. Of these, the a' to a pathway leads to a numbering that becomes discontiguous at position 32; the a to c pathway leads to a numbering discontiguity at position 36. Selection between the numberings of Figs. 25b and 25c, that have the first discontiguity at the same position (60), can be made considering that the numbering of Fig. 25b is built around an axis of higher order (Fu-3.2.1b) and is therefore preferred. The completion of the preferred discontiguous numbering is made according to Fu-3.2.4.

Fig. 25. Systematic numbering of (C₁₂₀-I_h)[4,6,10]fullerene