The sandpile is a deterministic diffusion process for configurations of chips (or grains) spreading out on a lattice. A nice high level introduction to the sandpile was written by Jordan Ellenberg.
Much of our work on the sanpdile concerns the single source sandpile. In this model, we begin with a stack of n chips at a single vertex of a lattice. The configuration evolves as follows: when the number of chips at a vertex is at least the degree of a vertex (4 for the square lattice), the vertex is eligible to topple; when a vertex topples, it loses a number of chips equal to its degree, and these chips are distributed (one each) to the neighbors of the vertex. The sandpile process evolves by toppling eligible vertices until no more such vertices exist, to produce a configuration in which every vertex has fewer chips than its degree. Surprisingly, this resulting configuration does not depend on the order in which topplings are carried out when there is more than one choice (the sandpile is Abelian).
In the interactive gallery below, you can see the striking final configurations on various lattices. The fractals and patterns in these configurations are the subject of the theorems in our papers on the sandpile.
Usage instructions are at the bottom of the gallery page.
In "Apollonian Structure in the Abelian Sandpile" (available here) with Lionel Levine and Charles Smart, we define a set Γ of stabilizeable 2×2 real symmetric matrices. In the paper, we conjecture a precise geometric structure for the closure of Γ, which allows us to construct fractal solutions to the Sandpile PDE which characterizes the continuum limit of the Abelian Sandpile. The purpose of this page is to present high resolution versions of computed images of the set Γ on various lattices.
In each case, there seems to be a discrete set P of "peak" matrices such that the closure of Gamma is the downset of P in the matrix order. In the case of the square lattice, we know the exact structure, but for the other lattices, we don't even have guesses.