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∈[smin,smax] 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 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 smin/smax→ 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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