Apollonian packings as physical fractals

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