Fullerenes having at least one symmetry axis (Cn, n>1) and a contiguous helical pathway (Fullerenes of this type without a contiguous helical pathway determined on the basis of Rules Fu-3.1.1 to Fu-3.1.4 are dealt with by Rules Fu-3.2.1 to Fu-3.2.4).
Rule Fu-3.1.1: Proper rotation axes (Cn) are examined in sequence from the highest-order to the lowest-order axis, until at least one contiguous helical pathway is found that begins in a ring through which a proper rotation axis passes, at one end of a bond bisected by a proper rotation axis, or at an atom through which a proper rotation axis passes. Numbering begins at the end of such a contiguous helical pathway, and the corresponding axis is called the "reference axis".
Rule Fu-3.1.2: If there is a choice for the selection of a reference axis or for the end of the reference axis to begin the numbering, a ring is preferred to a bond which, in turn, is preferred to an atom.
Rule Fu-3.1.2.1: When there is a choice between rings for the beginning of numbering, a larger ring is preferred to a smaller one. When there is still a choice, the preferred ring contains the highest-ranking atom at the first point of difference. The highest-ranking atom is defined as the common atom of the set of three largest rings. For example, in a [5,6]fullerene, an atom at the intersection of three six-membered rings (in short, a 6,6,6 atom) ranks higher than an atom at the intersection of two six-membered rings and a five-membered ring (a 6,6,5 atom).
Rule Fu-3.1.2.2: When there is a choice between bonds bisected by an axis for the beginning of numbering, the preferred bond contains the higher number of higher-ranking atoms.
Rule Fu-3.1.2.3: When there is a choice between atoms lying on an axis for the beginning of numbering, the numbering must start at the higher-ranking atom.
Rule Fu-3.1.3: When there is a choice among helical numbering pathways, the preferred pathway terminates as close as possible, in terms of the number of bonds, to the reference axis.
Rule Fu-3.1.4: When there is still a choice among helical pathways for numbering, they are evaluated by sequential comparison of the ranking of their atoms. The preferred pathway contains the highest-ranking atom at the first point of difference.
Fullerenes having at least one symmetry axis (Cn, n>1) but no contiguous helical numbering pathway determined on the basis of rules Fu-3.1.1 to Fu-3.1.4.
Rule Fu-3.2.1 (cf. Fu-3.1.1):
Rule Fu-3.2.3 (cf. Fu-3.1.4; for practical reasons, no rules analogous to Fu-3.1.3 is applied to segments of discontiguous numberings): When there is still a choice, the nth segments are compared according to rule Fu-3.1.4.
Once a preferred segment is identified based on rules Fu-3.2.1 to Fu-3.2.3, the numbering of the fullerene is completed by construction of the subsequent segment(s). The starting point of a new segment is defined according to rule Fu-3.2.4a (see below), and numbering within a segment is continued according to rule Fu-3.2.4b.
Rule Fu-3.2.4: After a discontiguity,
Rule Fu-3.3.1. Since there is no proper rotation axis, the plane of symmetry is used as the reference element. A search is made for a contiguous numbering helix that starts in a ring bisected by the plane of symmetry or along a bond lying in the plane.
Rule Fu-3.3.2. If there is a choice for the beginning of the numbering, rings bisected by the plane are preferred over bonds lying in the plane.
Rule Fu-3.3.2.1. If there is a choice among rings, a larger ring is preferred to a smaller one and, among rings of the same size, the preferred ring contains the highest-ranking atom at the first point of difference.
Rule Fu-3.3.2.2. If there is a choice among bonds lying in the plane, the preferred bond contains the higher number of higher-ranking atoms.
Rule Fu-3.3.3. If there is still a choice among contiguous numberings, the preferred pathway should:
Fullerenes belonging to the Ci or C1 point groups and having a contiguous helical numbering pathway.
Rule Fu-3.4.1. A larger ring is preferred to a smaller one to begin the numbering; among rings of the same size, the preferred ring contains the highest-ranking atom at the first point of difference.
Fu-3.4.2. If there is still a choice among contiguous numberings, they are compared according to their atom-ranking sequence. The preferred pathway contains the highest-ranking atom at the first point of difference.