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 $P$ and $Q$ of a vector space $V$, the Minkowski sum $P+Q$ is the set of points of the form $p+q$ for $p\in P$ and $q \in Q$. If $P$ and $Q$ are polyhedra, so is $P + Q$, and the vertices of $P+Q$ are sums of vertices of $P$ and $Q$. One natural way to think of $P+Q$ is that it is the projection of the product $P\times Q$ under the affine map $+:V\times V \to V$.

The simplest definition of a zonohedron (in any dimension) is that it is the Minkowski sum of finitely many intervals. Thus the faces of a zonohedra are themselves zonohedra. In 2 dimensions a zonohedron is a centrally symmetric polygon, and therefore has an even number of edges which come in parallel pairs of the same length. A zonohedron is convex, being the Minkowski sum of convex sets. Thus it is topologically a ball, and its boundary is topologically a sphere. A parallelepiped is an example of a 3-dimensional zonohedron; so is the rhombic dodecahedron and the rhombic triacontahedron. One can think of a zonohedron as a projection to a low dimensional space of a high dimensional parallelepiped; one can use this observation to produce interesting aperiodic tilings from zonohedra.

Here is Coxeter’s proof of the Sylvester-Gallai theorem. Let $Z= +_i I_i$ be a 3-dimensional zonohedron, expressed as the Minkowski sum of some collection of intervals $I_i$. Each $I_i$ determines a point $p_i$ in the projective plane; conversely, a collection of points in the projective plane determines a family of zonohedra, where each element of the family is determined by the edge lengths of the $I_i$. The faces of the zonohedra correspond to the colinear collections of $p_i$. A decomposition of the sphere into polygons meeting at least 3 to a vertex must contain at least one polygon with $<6$ sides, by Euler’s formula; hence every 3 dimensional zonohedron has at least one face with exactly $4$ sides. This corresponds to a line containing exactly 2 of the $p_i$; qed.