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I am a mathematician and not a chemist, and I am trying to understand the historical relationship and current usage of the word "fullerene" across Mathematics and Chemistry.

I apologise in advance if I say chemically-ridiculous things, but I am genuinely interested in understanding this.

I will assert some things that loosely describe my understanding of the situation, but which may be chemically wrong. Constructive correction is welcomed.

MY UNDERSTANDING FROM MATHEMATICS

(1) In graph theory (my area of Mathematics) a "fullerene graph" or simply "fullerene" is usually defined to be a 3-regular planar graph, where by "graph" we mean a combinatorial graph with vertices and edges (i.e. abstractions of atoms and bonds).

(2) The standard example of a fullerene is a 60-vertex planar graph, whose faces are arranged as in a standard soccer ball, and of course this is the famous geodesic dome structure or buckyball.

[Actually, mathematicians will also use the dodecahedron as a standard example of a fullerene graph because it has just 12 pentagonal faces and no hexagonal faces. This is fine for a mathematician because 0 hexagons is a perfectly sensible number of hexagons but perhaps it is chemically nonsensical.]

(3) This structure and its name arises from the carbon molecule $C_{60}$ which was first synthesised by Kroto et al., which has this spherical shape, and which they named buckminsterfullerene after Buckminster Fuller.

(4) Mathematics writings on fullerene graphs that attempt to explain the relationship to chemical fullerenes then usually say something like:

• an actual physical fullerene must have a molecular structure (atoms and bonds) that is a 3-valent planar graph with pentagonal and hexagonal faces




• there are lots of mathematically possible fullerenes, but only a minuscule fraction actually physically exist.




• to determine whether or not a mathematical structure might be physically realised, we compute the spectrum of the adjacency matrix and magically believe it relates to a mysterious chemistry thing called the HOMO-LUMO gap.

[In addition, some authors insist that to qualify as a fullerene graph, the pentagons must be separated from each other.]

[I guess that as carbon atoms are 4-valent, there must be a pile of double bonds floating around that us mathematicians have just totally ignored!]

MY QUESTIONS REGARDING THE CHEMISTRY

(1) When did the very specific term "buckminsterfullerene" get relaxed to "fullerene" and by whom?

(2) To exactly what class of carbon molecules does the term "fullerene" apply?

[I have learned that other forms of carbon include diamond where all atoms are 4-valent, then graphite, then fullerenes but I am hazy on the precise mathematical or chemical distinction. Then I am further confused by the fact that some sources distinguish tube structures (nano-tubes) from spherical structures and only call the latter fullerenes]

Sorry for the ridiculously long first question, but all clarification is gratefully received!

Fullerenes are distinct (meaning every fullerene type has a well defined structure) carbon-only molecules, where all atoms have three bonds (sp2, or nearly sp2, as they're not exactly planar).

Leaving out very freakish topologies (which are unlikely to be stable and not fullerenes), this leads to a closed shell-like structure where all surface elements are pentagons or hexagons. (Which are not perfect, bond lengths differ slightly if the edge is shared by two pentagons, two hexagons, or one-and-one.)

The simpler, smaller forms are spherical (C60) or prolate (C70, ...), larger ones can probably have a surface that is not completely convex. (anyone seen oblate fullerene structures?)

A carbon nanotube with a well defined, closed end structure can be a fullerene, but they usually haven't got one.

The actual word "fullerene" comes from the geodesic dome designed by Buckminster Fuller. The near-spherical structure of $ce{C_{60}}$, the simplest and best known of these molecules, is the dual of the triangular-faced geodesic dome. Mathematical definitions of "fullerenes" come from the same source.