## More on Buckyballs

Google has a nice doodle celebrating the 25th anniversary of the **buckyball**. (A video of the doodle can be seen here.)

“Buckyball” is the informal name of a particular kind of carbon molecule that, geometrically, resembles the geodesic dome made popular by futurist Buckminster Fuller. They are more generally known as **fullerenes** (again, after Fuller), and among other things, have recently been detected in space.

Viewed mathematically/geometrically/graph-theoretically, a **fullerene **is a solid consisting of only pentagonal and hexagonal faces. There are many different fullerenes–for example, having 20, 70, or 200 vertices–but what’s amazing is that apparently all of them have **exactly 12 pentagonal faces**. Only the number of **hexagonal** faces changes.

Apparently this fact is a direct consequence of **Euler’s formula**, namely *V – E + F *= 2, where *V*, *E*, and *F* are the number of vertices, edges, and faces, respectively, in a given solid. For example, a **cube** has 8 **vertices**, 12 **edges**, and 6 **faces**; note that** **8 – 12 + 6 = 2, just as Euler requires.

Try verifying Euler’s formula for an octahedron! Then, when you’re done with that, prove the above remark about fullerenes.

*Click here to see more in Geometry.*