Estimate for the fractal dimension of the Apollonian gasket in d dimensions

TL;DR

This paper adapts a recent theory to estimate the Hausdorff dimension of the Apollonian gasket across multiple dimensions, providing accurate predictions in 2D and 3D and extending estimates to higher dimensions.

Contribution

It introduces a novel adaptation of a theory for sphere packings to predict the fractal dimension of Apollonian gaskets in various dimensions.

Findings

Predicted Hausdorff dimension matches known values in 2D and 3D within 0.05% and 0.6%.

Provides new estimates for dimensions 4 to 8.

Demonstrates the theory's applicability to higher-dimensional fractals.

Abstract

We adapt a recent theory for the random close packing of polydisperse spheres in three dimensions [R. S. Farr and R. D. Groot, J. Chem. Phys. {\bf 131} 244104 (2009)] in order to predict the Hausdorff dimension $d_{A}$ of the Apollonian gasket in dimensions 2 and above. Our approximate results agree with published values in $2$ and $3$ dimensions to within $0.05%$ and $0.6%$ respectively, and we provide predictions for dimensions $4$ to $8$.