Leonhard Euler (1707-1783)

Euler showed that for all simple polyhedra the Euler characteristic is C=2. As an example let's look at a cube: it has V=8 vertices, E=12 edges, and F=6 faces, hence C=8-12+6=2. Do you get the same for, say, a tetrahedron? The hard work, of course, is to show that C is an invariant for all simple polyhedra.

For the special case of a simple polyhedron made from only pentagons and hexagons, the Euler formula (1) simplifies considerably. Let P be the number of pentagons and H be the number of hexagons. Clearly the number of faces is then F=P+H. Each edge is shared by two polygons, so the number of edges is E=(5 P+6 H)/2. The number of vertices is a bit more tricky. There must be at least three polygons sharing a vertex. On the other hand, for a convex polyhedron the sum of the internal angles of the polygons must not exceed 360 degrees. Since the internal angle for a pentagon (hexagon) is 108 (120) degrees, there must not be more than three polygons sharing a vertex. Thus V=(5 P+6 H)/3. Inserting all this into (1) and using C=2, we end up with

I.e. a simple polyhedron entirely made from pentagons and hexagons must contain exactly 12 pentagons. The number of hexagons can be arbitrary. Given the number of hexagons, we can now easily calculate the number of vertices:

But for a fullerene the number of vertices is just the number of carbon atoms in the molecule. We have thus found that fullerenes must have an even number of carbon atoms, and that there is a lower limit for the size of the fullerenes: C20 is the smalles fullerene. It looks like a dodecahedron.

If you want to learn more about Leonhard Euler and his theorem

• have a look at his biography from the class Concepts in Physics, Notions in Art at the University of California, Los Angeles
• read his biography on the History of Mathematics site at the University of St Andrews
• see fifteen proofs of Eulers Formula in the The Geometry Junkyard