The Amazing Buckyball

3. The Buckyball Asteroid

The Amazing Buckyball

One of the shapes you saw on the space-capsule screen (see "The Buckyball Asteroid") is a truncated icosahedron.

Truncated icosahedron.

A regular icosahedron is made up of twenty equilateral triangles that meet at twelve points, or vertices. To create a truncated icosahedron, you chop off each vertex so the twelve vertices turn into twelve regular pentagons and the twenty equilateral triangles become twenty regular hexagons.

Lopping off one corner of an icosahedron (above) is the first step in creating a truncated icosahedron (below).

The resulting shape has twenty hexagons and twelve pentagons as its surface, thirty-two faces in all.

Pattern showing all thirty-two faces for constructing your own truncated icosahedron (buckyball).

Three-dimensional geometry is very useful for describing the arrangement of atoms in different materials. Carbon atoms, for example, can arrange themselves in a pattern that looks a lot like neatly stacked tetrahedra. Arranged in this way, carbon atoms form diamonds, one of the hardest materials known.

Tetrahedral arrangement of carbon atoms in diamond.

In contrast, when carbon atoms are arranged in hexagonal rings linked together into to form vast sheets, they form graphite, a soft material used in lubricants and in pencils. The hexagonal pattern looks like a honeycomb grid (see "Paving the Plane").

In the 1980s, scientists discovered carbon molecules in the shape of truncated icosahedra. Each molecule consists of sixty carbon atoms. They named the molecule buckminsterfullerene, after R. Buckminster Fuller, the engineer, mathematician, and architect who had studied and designed buildings with a similar structure. Scientists sometimes call these molecules "buckyballs" for short.

Computer-generated model of a buckyball molecule, which consists of sixty carbon atoms arranged to form a spherical cage.

If you count the number of hexagons and pentagons on the surface of a classic soccer ball, you will realize that when you play soccer, you are kicking a truncated 

icosahedron, or buckyball!

Classic soccer ball.

Actually, it's not quite a truncated icosahedron, because the pentagons and hexagons on a true mathematical buckyball are flat. On a soccer ball, they are rounded so that the ball is essentially spherical.

TRY IT!

Show that Euler's Rule (see "Solid Faces") for the number of faces, vertices, and edges on a polyhedron works for a truncated icosahedron (buckyball).

Answers:

A soccer ball has 32 faces (20 hexagons and 12 pentagons), so F = 32.

A soccer ball is akin to a truncated icosahedron, so its 12 pentagons come from "slicing off" the 12 vertices of an icosahedron. Therefore, a truncated icosahedron has 12 ✕ 5 = 60 vertices.

F + V − E = 32 + 60 −  E = 2. Solving the equation gives 90 as the number of edges.

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