Circles in the Sand
I will describe the role played by an Apollonian circle packing in the
scaling limit of the abelian sandpile on the square grid \(Z^2\). The
sandpile solves a certain integer optimization problem. Associated to
each circle in the packing is a locally optimal solution to that
problem. Each locally optimal solution can be described by an infinite
periodic pattern of sand, and the patterns associated to any four
mutually tangent circles obey an analogue of the Descartes Circle
Theorem. Joint work with Wesley Pegden and Charles Smart.