Apollonian packings as physical fractals

Details Apollonian packings as physical fractals Apollonian packings as physical fractals

2011-11-22

arxiv.org/abs/1111.5256v1

Molecular Physics, Vol. 109, Iss. 22, 2011

Physics

Condensed Matter

Statistical Mechanics

6 pages, 3 figures, 1 table

Scientific paper

10.1080/00268976.2011.630598

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 which generates random APs of spheres with a finite range of diameters: the correct asymptotic limit $s_\mathrm{min}/s_\mathrm{max}\rightarrow 0$ and the known APs' fractal dimensions are recovered and an excellent agreement with the generalized analytic laws is proved within the overall ranges of sizes.

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