Single line Apollonian gaskets: is the limit a space filling fractal curve?

TL;DR

This paper investigates whether a single-line, nested approximation of the Apollonian gasket converges to a space-filling fractal curve, offering a novel perspective on fractal geometry and curve representation.

Contribution

It introduces a new method of representing Apollonian gaskets as continuous, non-crossing lines and explores their potential to form space-filling fractal curves.

Findings

Single-line approximations can approach space-filling fractals.

Nested configurations influence the fractal properties of the line.

The limit curve's space-filling nature is analyzed through these approximations.

Abstract

In this manuscript we study single-line approximations and fractals based on the Apollonian gasket. The well-known Apollonian gasket is the limit case of configurations of kissing circles. Rather than plotting the circles as discs on a differently colored background (the traditional representation), we draw all circles as one line without lifting the pen and without crossing itself. Moreover, the configurations are nested. In this manuscript we explore whether the limit of the line drawings gives rise to a space filling fractal curve.