Estimate for the fractal dimension of the Apollonian gasket in d dimensions
Techniques from random sphere packing predict the dimension of the Apollonian gasket, a fractal made up of non-overlapping hyperspheres.
Physical Review E 81, 61403 (2010)
R. Farr, E. Griffiths
















We adapt a recent theory for the random close packing of polydisperse spheres in three dimensions in order to predict the Hausdorff dimension 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.
More in Puzzles in packing
Shear elastic deformation in cells
Analysis of the linear elastic behaviour of plant cell dispersions improves our understanding of how to stabilise and texturise food products.
Viscosity of polydisperse spheres
A quick and simple way to evaluate the packing fraction of polydisperse spheres, which is a measure of how they crowd around each other.
Random close packing fractions
Lognormal distributions (and mixtures of same) are a useful model for the size distribution in emulsions and sediments.