The fullerenes generator

Fullerenes are spherically shaped molecules built entirely from carbon atoms. Each of these has bonds to exactly three other carbon atoms. The carbon atoms form rings of either five atoms (pentagons) or six atoms (hexagons). Due to the Euler formula there are exactly twelve pentagons. Our fullerene generator is called "fullgen". For a scientific article on fullgen, see the bottom of this page.)

The best-known fullerene is the "buckyball" molecule which consists of 60 carbon atoms. If you try running fullgen with the number of atoms set to 60, you will find that there are far more C₆₀ fullerenes than just the "buckyball" -- 1812 differently built molecules actually. The "buckyball" molecule stands out from the full set by its "football" look, which is a very symmetric shape -- some authors have called the "buckyball" the most symmetric molecule in the world. In mathematical terms, it has a large symmetry group, i.e. there are many different rotations and reflections which, applied to the buckyball, will permutate the atoms but give you the same shape in the same position. There are two ways of reducing fullgen's output set to just the "buckyball" when C₆₀ fullerenes are generated.

There is a button in the fullgen options window that deals with symmetries, it is labelled "Symmetry Filter". Clicking it will bring up a new dialogue that presents you with 24 different symmetry groups, each named with a symbolic abbreviation used in chemistry. The buckyball's symmetry group is "icosahedral symmetry", known as Ih. When the symmetry filter dialogue is brought up for the first time, all symmetry groups are selected (there is a button for each symmetry in the dialogue, and each of these will be pressed). By clicking the "Clear all" button and then the "Ih" button in the symmetry filter dialogue, the filter is changed and becomes active. Fullgen will still generate all 1812 C₆₀ fullerenes, but the symmetry filter subroutine will discard every molecule whose symmetry group is not Ih, and so just one molecule -- the buckyball -- will be written to fullgen's output.

The buckyball has another feature of interest in chemistry. As mentioned before, its rings are connected just like a football is stitched together from hexagonal and pentagonal leather patches. (That's soccer football in case you were wondering.) Looking at a (black) pentagon patch on such a football, you willl see that it is surrounded by (white) hexagon patches, not touching any other pentagon. Mathematical chemists say that the football follows the "isolated pentagon rule" (ipr). You can restrict fullgen to generate only molecules conforming to ipr by clicking the respective checkbox in its options window. Unlike the symmetry filter, the ipr option will affect fullgen's operation at an early stage so that generation time will be significantly smaller: fullgen won't generate all 1812 C₆₀ fullerenes if it knows that only the subset defined by ipr is requested. It turns out that the buckyball is the only ipr-compliant C₆₀ isomer.

Another option for fullgen will cause it to output the dual of every graph it has generated. As fullerenes are 3-regular graphs (each atom has bonds to three other atoms), the dual graph is a triangulation (all its faces are triangles).

Fullerenes have become immensely popular, and their discovery was rewarded with the 1996 Nobel Prize in chemistry. Here are some links. (Each will open in a new browser window.)

The fullerenes generator was written by Gunnar Brinkmann; the underlying mathematical research was a collaboration with Andreas Dress. For more information and as a scientific reference, see