Apollonian packings as physical fractals

Abstract

The Apollonian packings (APs) are fractals that result from a space-filling procedure with spheres. We discuss the finite size effects for finite intervals $s\in [s_{\min },s_{\max }]$ 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 $s_{\min }/s_{\max }\to 0$ 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.

6 pages, 3 figures, 1 table

References in corpus (1)

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

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