Apollonian packings as physical fractals

TL;DR

This paper introduces a generalized analytical framework for finite-size effects in Apollonian packings, supported by a new algorithm that accurately reproduces their fractal properties across size ranges.

Contribution

It provides the first analytical generalization of scale-free laws for finite-size Apollonian packings and validates it with a novel space-filling algorithm.

Findings

Analytical laws accurately describe finite-size effects.

Algorithm successfully generates random APs with finite size ranges.

Results match known fractal dimensions and asymptotic limits.

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 $s\in[s_\mathrm{min},s_\mathrm{max}]$ 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.