The Geometry of the Buckyball

The icosahedron is one of the five Platonic solids (3-D shapes whose faces are identical regular polygons. The other four are the tetrahedron, the cube, the octahedron, and the dodecahedron). It has 20 faces, each one an equilateral triangle. Chopping off each vertex reveals the 12 pentagonal and 20 hexagonal faces of the truncated icosahedron, which is one of the 13 Archimedean solids (shapes made from truncating Platonic solids in certain ways). The process is shown below:

This object is highly symmetric, and has quite an interesting topology. For details, I refer you to an excellent article by Fan Chung and Shlomo Sternberg called "Mathematics and the Buckyball" (American Scientist vol. 81, page 56). The figures on this page come from their paper, which unfortunately is not available on-line yet.

Make your own Buckyball!

Obviously, buckyballs in nature are not made by truncating an icosahedron. In fact, they are made when a graphite sheet "rolls up" and changes a few hexagons into pentagons in order to reduce its energy. The process involves striking an arc between two graphite electrodes in the presence of a helium atmosphere, which produces all kinds of fullerenes, not just ones with 60 atoms. By the time you strain out all the carbon garbage and heavier fullerenes, the buckyballs left constitute a mere 2-3% of the total mass. This is why C₆₀ powder of 99.99% purity costs upward of $150/gram. (Or at least it did in the late 1990's; I don't know the current market price).

But you can make your own model of a buckyball for much less than that. Just print out the figure below, cut out the hexagons, and start folding along the lines common to 2 hexagons. You will find that the flat sheet neatly curls up into a sphere-like object as rings of hexagons are connected by pentagons (really, this is easy! The figure pratically makes itself once you start folding). This clearly illustrates how a graphite sheet rolls up into a buckyball-- a few pieces of tape, and you'll have your own truncated icosahedron. Enjoy!

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Email: kimall (at symbol) mindspring.com