Mathematics

"Euler's formula for polyhedra"

Consider a cube: there are eight vertices (corners), twelve edges (lines connecting vertices) and six "faces" (two-dimensional surfaces bounded by edges). Compute the no. of vertices minus no. of edges plus no. of faces. We get 8 - 12 + 6 = 2.

Let's look at another figure, say a square-based pyramid. Again we get V - E + F = 5 - 8 + 5 = 2.

This formula, that V - E + F = 2, is known as Euler's formula, or Euler characteristic, for polyhedra. Nowadays many different proofs of this formula exist. One website lists nineteen proofs. Why not be satisfied with one proof? After all, one proof is enough to know that it is true.

Sometimes another proof is instructive because it uses another approach. Some proofs can be adapted easier to prove other properties. Some proofs are short and efficient, but do not provide a picture of what is going on. Others are very descriptive, but longer. One of the proofs show that if one assumes that Euler's characteristic holds for any polyhedron with a certain number "n" of vertices, then the result also holds for "n+1" vertices. So one only
needs to prove it for a polyhedron with one vertex, to get the result for any number of vertices. (This is like saying that if an infinite sequence of dominoes are close enough, pushing the first one will lead to all of the dominoes falling.)

Other proofs focus on the edges, or faces, rather than the vertices. There is a proof that starts by imagining a "+" charge at every vertex, and a "-" charge in the middle of every edge, and then goes on to show that they all cancel out except for two positive charges.

Many of the proofs can be illustrated by a graphical animation. If so, we will learn to write a computer program to do the animation.

In this project, we will examine, debate, and illustrate many different proofs of Euler's
formula. Each new proof will teach us a new piece of mathematics.

Participants

• Mr. Gusti van Zyl