The interactive gallery below let you browse and zoom very large sandpiles on various lattices.
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.