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

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 $8$.Comment: 4 pages, 4 figure