3.3 KLs and fullerenes
Among the elements, carbon C is the basis of all life. A whole branch of chemistry, organic chemistry, is devoted to the study of C-C bonds and different molecules originating from them. Carbon is the only 4-valent element able to produce long homoatomic stable chains or different 4-valent nets. The other 4-valent candidate for this might be only silicon Si, with its homoatomic chemistry is only beginning to be developed.
In addition to long-known diamond and graphite, a new form of carbon, the fullerene C60, was first synthesized by H.W. Kroto, R.F. Curl and R.E. Smalley in 1985. Along with its structure, that of a spherical closed pentagonal/hexagonal monoatomic shell, it has the remarkable property of rotational symmetry of order 5 (which according to Barlow's crystallographic restriction theorem is forbidden in crystallographic space or plane symmetry groups) and the highest possible, icosahedral, point-group symmetry. Since the discovery of C60, different fullerenes (e.g., C70, C76, C78, C82, C84, etc.) have been synthesized, opening a new field for research of potentially different possible fullerene structures from point of view of geometry, graph theory, or topology. The most complete discussion of fullerenes is given by P.D. Fowler and D.E. Manolopoulos (1995).