ID: 1006.1606

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

June 8, 2010

View on ArXiv

Similar papers 2

Spatial Statistics of Apollonian Gaskets

May 17, 2017

83% Match
Weiru Chen, Mo Jiao, Calvin Kessler, ... , Zhang Xin
Metric Geometry
Dynamical Systems
Mathematical Physics

Apollonian gaskets are formed by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We experimentally study the pair correlation, electrostatic energy, and nearest neighbor spacing of centers of circles from Apollonian gaskets. Even though the centers of these circles are not uniformly distributed in any `ambient' space, after proper normalization, all these statistics seem to exhibit some interesting limiting behaviors.

Find SimilarView on arXiv

Comment on "Explicit Analytical Solution for Random Close Packing in d=2 and d=3", Physical Review Letters {\bf 128}, 028002 (2022)

January 25, 2022

83% Match
Raphael Blumenfeld
Disordered Systems and Neura...
Soft Condensed Matter
Mathematical Physics

The method, proposed in \cite{Za22} to derive the densest packing fraction of random disc and sphere packings, is shown to yield in two dimensions too high a value that (i) violates the very assumption underlying the method and (ii) corresponds to a high degree of structural order. The claim that the obtained value is supported by a specific simulation is shown to be unfounded. One source of the error is pointed out.

Find SimilarView on arXiv

Exact analytical solution of average path length for Apollonian networks

June 24, 2007

82% Match
Zhongzhi Zhang, Lichao Chen, Shuigeng Zhou, Lujun Fang, ... , Zou Tao
Statistical Mechanics
Physics and Society

The exact formula for the average path length of Apollonian networks is found. With the help of recursion relations derived from the self-similar structure, we obtain the exact solution of average path length, $\bar{d}_t$, for Apollonian networks. In contrast to the well-known numerical result $\bar{d}_t \propto (\ln N_t)^{3/4}$ [Phys. Rev. Lett. \textbf{94}, 018702 (2005)], our rigorous solution shows that the average path length grows logarithmically as $\bar{d}_t \propto \...

Find SimilarView on arXiv

Self-similar disk packings as model spatial scale-free networks

July 29, 2004

82% Match
Jonathan P. K. Doye, Claire P. Massen
Statistical Mechanics

The network of contacts in space-filling disk packings, such as the Apollonian packing, are examined. These networks provide an interesting example of spatial scale-free networks, where the topology reflects the broad distribution of disk areas. A wide variety of topological and spatial properties of these systems are characterized. Their potential as models for networks of connected minima on energy landscapes is discussed.

Find SimilarView on arXiv

The Residual Set Dimension of a Generalized Apollonian Packing

October 16, 2023

82% Match
Daniel Lautzenheiser
Number Theory
Metric Geometry

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.

Find SimilarView on arXiv

Statistical mechanics of the lattice sphere packing problem

May 6, 2013

82% Match
Yoav Kallus
Statistical Mechanics
Metric Geometry
Mathematical Physics

We present an efficient Monte Carlo method for the lattice sphere packing problem in d dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond previous methods not only in exploring higher dimensions but also in shedding light on the statistical mechanics underlying the problem in question. We observe evidence of a phase transition in the thermodynamic limit $d\to\infty$. In the dimen...

Find SimilarView on arXiv

Box counting dimensions of generalised fractal nests

February 2, 2018

82% Match
Siniša Miličić
Metric Geometry

Fractal nests are sets defined as unions of unit $n$-spheres scaled by a sequence of $k^{-\alpha}$ for some $\alpha>0$. In this article we generalise the concept to subsets of such spheres and find the formulas for their box counting dimensions. We introduce some novel classes of parameterised fractal nests and apply these results to compute the dimensions with respect to these parameters. We also show that these dimensions can be seen numerically. These results motivate furt...

Find SimilarView on arXiv

Energy landscapes, scale-free networks and Apollonian packings

December 6, 2006

82% Match
Jonathan P. K. Doye, Claire P. Massen
Statistical Mechanics

We review recent results on the topological properties of two spatial scale-free networks, the inherent structure and Apollonian networks. The similarities between these two types of network suggest an explanation for the scale-free character of the inherent structure networks. Namely, that the energy landscape can be viewed as a fractal packing of basins of attraction.

Find SimilarView on arXiv

Random perfect lattices and the sphere packing problem

February 25, 2012

82% Match
Alexei Andreanov, Antonello Scardicchio
Statistical Mechanics

Motivated by the search for best lattice sphere packings in Euclidean spaces of large dimensions we study randomly generated perfect lattices in moderately large dimensions (up to d=19 included). Perfect lattices are relevant in the solution of the problem of lattice sphere packing, because the best lattice packing is a perfect lattice and because they can be generated easily by an algorithm. Their number however grows super-exponentially with the dimension so to get an idea ...

Find SimilarView on arXiv

Exactly Solvable Disordered Sphere-Packing Model in Arbitrary-Dimension Euclidean Spaces

March 11, 2006

82% Match
S. Torquato, F. H. Stillinger
Soft Condensed Matter
Statistical Mechanics

We introduce a generalization of the well-known random sequential addition (RSA) process for hard spheres in $d$-dimensional Euclidean space $\mathbb{R}^d$. We show that all of the $n$-particle correlation functions of this nonequilibrium model, in a certain limit called the "ghost" RSA packing, can be obtained analytically for all allowable densities and in any dimension. This represents the first exactly solvable disordered sphere-packing model in arbitrary dimension. The f...

Find SimilarView on arXiv