The deep locus of a cluster algebra A consists of the complement to all the cluster tori inside Spec(A). It is a challenging problem to determine when it is nonempty, partly because there are usually an infinite number of cluster tori. By passing to finite fields, one can hope to obtain a finite number of cluster tori covering the complement of the deep locus, and lift these to characteristic zero. While this turns out not to be the case, studying cluster tori over finite fields leads to interesting combinatorics. In particular, we will see how to recover from a cluster-theoretic perspective the "hexagonal edges" of the universal polytope of all partitions of an n-gon. This is based in joint works with different subsets of {Marco Castronovo, Mikhail Gorsky, Daniel Pérez Melesio, David Speyer}, and I won't assume prior knowledge of cluster algebras.