Any hollow, cage-like molecule consisting entirely of carbon atoms in a state of sp2 hybridization and composed of five- and six-membered rings is referred to as a fullerene [124]. This class of carbon nanostructures thus comprises the quasi-spherical species that are popularly known as buckyballs as well as axially confined carbon nanotubes, among other systems. As every carbon atom on the fullerene surface is threefold coordinated, a fullerene may be understood as a rolled-up graphene sheet, as illustrated in Figure 6.1 by the example of a buckyball. Euler's theorem, applied to polyhedra that consist exclusively of hexagons and pentagons, confines the number of pentagons to twelve [125], while there may be an arbitary number of hexagons, excepting a single one [126]. This implies that the smallest fullerene of this type is a combination of twelve pentagons, C20. Quasi-fullerenes contain rings beyond pentagons and hexagons, such as heptagons [127] or octagons [128]. The effect of these structural irregularities is a major enhancement of fullerene reactivity.

C60

The most readily available fullerene species, and the prototypical molecular allotrope of carbon, is C60, also known as Buckminster fullerene, making fully explicit the name patron of the fullerenes, the American architect, author and visionary Richard Buckminster Fuller (1895–1983), whose name is famously associated with designs based on geodesic dome structures. The C atoms of C60 are positioned at the vertices of a regular truncated icosahedron. The point group of C60 is Ih, the number of symmetry operations of the molecule, i.e. operations that map C60 into itself, is 120.

Fullerenes satisfy the Isolated Pentagon rule, stating that structures with pentagons entirely surrounded by hexagons are more stable than those that admit pentagon adjacency. The geometry of C60, where all pentagons are isolated is unique among the fullerenes Cn with n ≤ 70. For topological reasons, all other species of this class involve adjacent pentagons. While pure cages containing these substructures are deemed too unstable to be fabricated in the laboratory [129], they can be stabilized by external or internal impurities (e.g. [130, 131]).

The average bond length between the atoms of C60 is 1.44 Å [132]. Each atom forms two single bonds and one double bond with adjacent atoms.