Computing the Hausdorff dimension of the Apollonian gasket

Abstract

The Apollonian gasket is a well-studied circle packing, created by iteratively filling a region with mutually tangent circles. Important properties of the packing, including the distribution of the circle radii, are universal and governed by its Hausdorff dimension. No closed form is currently known for the Hausdorff dimension, and numerically estimating it is a special case of a more general and hard problem: effective, rigorous estimates of dimension of a limit set generated by non-uniform contractions. In this talk, I will talk about an approach that can efficiently solve this problem. With this method we can not only compute the dimension of the gasket to a surplus of decimal places, but also rigorously validate this computation as a mathematical theorem. Our method is not particularly specialised to the Apollonian gasket, and could generalise easily to other “difficult" parabolic fractals.