Fullerene

Fullerenes are one of only four types of naturally occurring forms of carbon (the other three being diamond, graphite and ceraphite). They are molecules composed entirely of carbon, taking the form of a hollow sphere, ellipsoid, or tube. Spherical fullerenes are sometimes called buckyballs, while cylindrical fullerenes are called buckytubes or nanotubes.

Fullerenes in mathematics

In mathematics, a fullerene is a trivalent

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convex polyhedron with pentagonal and hexagonal faces.

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Using Euler formula, one can easily prove that there are

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exactly 12 pentagons in a fullerene. The smallest fullerene is C20, the dodecahedron. There are

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no fullerenes with 22 vertices. The number of fullerenes

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C2n grows rapidly with increasing n = 12,13, ... For instance, there are 1812 non-isomorphic

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fullerenes C60 but only one of them, the buckminsterfullerene alias truncated icosahedron, has no pair of adjacent pentagons.

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