Zonohedra and the Sylvester-Gallai theorem

When I was in Melbourne recently, I spent some time browsing through a copy of “Twelve Geometric Essays” by Harold Coxeter in the (small) library at AMSI. One of these essays was entitled “The classification of zonohedra by means of projective diagrams”, and it contained a very cute proof of the Sylvester-Gallai theorem, which I thought would make a nice (short!) blog post.

The Sylvester-Gallai theorem says that a finite collection of points in a projective plane are either all on a line, or else there is some line that contains exactly two of the points. Coxeter’s proof of this theorem falls out incidentally from an apparently unrelated study of certain polyhedra known as zonohedra.

For subsets and of a vector space , the Minkowski sum  is the set of points of the form for and . If and are polyhedra, so is , and the vertices of are sums of vertices of and . One natural way to think of is that it is the projection of the product under the affine map .