The Abelian Sandpile is a diffusion process on configurations of chips on the integer lattice $\Z2$; a vertex with at least 4 chips \emph{topples}, distributing one chip to each of its neighbors. One of the most striking unexplained features of the sandpile is that it appears to produce terminal configurations converging to a peculiar fractal limit when begun from increasingly large stacks of chips at the origin. In this talk, we will discuss a mathematical explanation for this fractal behavior. We will present a conjecture regarding a precise fractal structure based on Apollonian circle packings for a certain class of 2x2 matrices, which allows us to construct fractal solutions giving exact geometric descriptions of portions of the limiting sandpile