Appendix 1 - Rules

Fullerenes having at least one symmetry axis (C_n, 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 (C_n) 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 (C_n, 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):

1. the nth discontiguity must occur as late as possible in the numbering.

2. among different pathways having the nth discontiguity at the same highest locant, the preferred numbering pathway is built around the proper rotation axis of higher order.

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,

1. the next atom to be numbered must be directly connected to one of the previously numbered atoms; among the latter, the atom with the highest locant is preferred.

2. numbering continues by the shortest path, in terms of number of bonds, to reach an unnumbered atom that is directly connected to a numbered atom with the lowest locant already assigned. (In this way the sense of the numbering helix should be maintained, and its pitch kept as small as possible.)

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:

1. begin, and

2. terminate as close as possible to the reference plane in terms of number of bonds.

Fullerenes belonging to the C_i 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.