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A couple of months ago I discussed a method to reduce a dynamical problem (computing the maximal rotation number of a prescribed element $w$ in a free group given the rotation numbers of the generators) to a purely combinatorial one. Now Alden Walker and I have uploaded our paper, entitled “Ziggurats and rotation numbers”, to the arXiv.

The purpose of this blog post (aside from continuing the trend of posts titles containing the letter “Z”) is to discuss a very interesting conjecture that arose in the course of writing this paper. The conjecture does not need many prerequisites to appreciate or to attack, and it is my hope that some smart undergrad somewhere will crack it. The context is as follows.

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 $+:V\times V \to V$.