A game of packings

TL;DR

This paper explores a family of circle packings with the Apollonian property, generated from initial configurations of three mutually tangent circles, revealing new structures within infinite packings.

Contribution

It introduces a novel infinite family of circle packings with the Apollonian property derived from a simple initial configuration.

Findings

Generated infinite circle packings with the Apollonian property.

Identified structural properties of these packings.

Provided a framework for constructing new packings from initial configurations.

Abstract

In this note, we investigate an infinite one parameter family of circle packings, each with a set of three mutually tangent circles. We use these to generate an infinite set of circle packings with the Apollonian property. That is, every circle in the packing is a member of a cluster of four mutually tangent circles.