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

The Apollonian gasket is a well-studied circle packing. Important properties of the packing, including the distribution of the circle radii, are governed by its Hausdorff dimension. No closed form is currently known for the Hausdorff dimension, and its computation is a special case of a more general and hard problem: effective, rigorous estimates of dimension of a parabolic limit set. In this paper we develop an efficient method for solving this problem which allows us to com...

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 revisit the scaling properties of growing spheres randomly seeded in d=2,3 and 4 dimensions using a mean-field approach. We model the insertion probability without assuming a priori a functional form for the radius distribution. The functional form of the insertion probability shows an unprecedented agreement with numerical simulations in d=2, 3 and 4 dimensions. We infer from the insertion probability the scaling behavior of the Random Apollonian Packing and its fractal d...

The fractal properties of models of randomly placed $n$-dimensional spheres ($n$=1,2,3) are studied using standard techniques for calculating fractal dimensions in empirical data (the box counting and Minkowski-sausage techniques). Using analytical and numerical calculations it is shown that in the regime of low volume fraction occupied by the spheres, apparent fractal behavior is observed for a range of scales between physically relevant cut-offs. The width of this range, ty...

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 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.

We present the first study of disordered jammed hard-sphere packings in four-, five- and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions $d=4$, 5 and 6 to be $\phi_{MRJ} \simeq 0.46$, 0.31 and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form $\phi_{MRJ}= c_1/2^d+(c_2 d)/2^d$, where ...

We propose a simple algorithm which produces a new category of networks, high dimensional random Apollonian networks, with small-world and scale-free characteristics. We derive analytical expressions for their degree distributions and clustering coefficients which are determined by the dimension of the network. The values obtained for these parameters are in good agreement with simulation results and comparable to those coming from real networks. We prove also analitically th...

In this short commentary we provide our comment on the article "Explicit Analytical Solution for Random Close Packing in $d=2$ and $d=3$" and its subsequent Erratum that are recently published in Physical Review Letters. In that Letter, the author presented an explicit analytical derivation of the volume fractions $\phi_{\rm RCP}$ for random close packings (RCP) in both $d=2$ and $d=3$. Here we first briefly show the key parts of the derivation in Ref.~\cite{Za22}, and then p...

Employing numerical and theoretical methods, we investigate the structural characteristics of random sequential addition (RSA) of congruent spheres in $d$-dimensional Euclidean space $\mathbb{R}^d$ in the infinite-time or saturation limit for the first six space dimensions ($1 \le d \le 6$). Specifically, we determine the saturation density, pair correlation function, cumulative coordination number and the structure factor in each =of these dimensions. We find that for $2 \le...