Hausdorff dimension of the Apollonian gasket

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 $8$.

In this manuscript we study single-line approximations and fractals based on the Apollonian gasket. The well-known Apollonian gasket is the limit case of configurations of kissing circles. Rather than plotting the circles as discs on a differently colored background (the traditional representation), we draw all circles as one line without lifting the pen and without crossing itself. Moreover, the configurations are nested. In this manuscript we explore whether the limit of th...

We view space-filling circle packings as subsets of the boundary of hyperbolic space subject to symmetry conditions based on a discrete group of isometries. This allows for the application of counting methods which admit rigorous upper and lower bounds on the Hausdorff dimension of the residual set of a generalized Apollonian circle packing. This dimension (which also coincides with a critical exponent) is strictly greater than that of the Apollonian packing.

Apollonian gaskets are formed by repeatedly filling the gaps between three mutually tangent circles with further tangent circles. In this paper we give explicit formulas for the the limiting pair correlation and the limiting nearest neighbor spacing of centers of circles from a fixed Apollonian gasket. These are corollaries of the convergence of moments that we prove. The input from ergodic theory is an extension of Mohammadi-Oh's Theorem on the equidisribution of expanding h...

The Apollonian packings (APs) are fractals that result from a space-filling procedure with spheres. We discuss the finite size effects for finite intervals $s\in[s_\mathrm{min},s_\mathrm{max}]$ between the largest and the smallest sizes of the filling spheres. We derive a simple analytical generalization of the scale-free laws, which allows a quantitative study of such \textit{physical fractals}. To test our result, a new efficient space-filling algorithm has been developed w...

We give conditions for the existence of the Minkowski content of limit sets stemming from infinite conformal graph directed systems. As an application we obtain Minkowski measurability of Apollonian gaskets, provide explicit formulae of the Minkowski content, and prove the analytic dependence on the initial circles. Further, we are able to link the fractal Euler characteristic, as well as the Minkowski content, of Apollonian gaskets with the asymptotic behaviour of the circle...

We propose a simple algorithm which produces high dimensional Apollonian networks with both small-world and scale-free characteristics. We derive analytical expressions for the degree distribution, the clustering coefficient and the diameter of the networks, which are determined by their dimension.

A construction and algebraic characterization of two unbounded Apollonian Disk packings in the plane and the half-plane are presented. Both turn out to involve the golden ratio.

The curvatures of the circles in integral Apollonian circle packings, named for Apollonius of Perga (262-190 BC), form an infinite collection of integers whose Diophantine properties have recently seen a surge in interest. Here, we give a new description of Apollonian circle packings built upon the study of the collection of bases of Z[i]^2, inspired by, and intimately related to, the `sensual quadratic form' of Conway.

In this note, we give an elementary proof that the Hausdorff dimension of the Rauzy gasket is at most 1.7407, improving upon results of Avila et al. and Fougeron.