Apollonian circle packings of the half-plane

TL;DR

This paper characterizes when Apollonian circle packings of a half-plane are related by Euclidean similarities, linking their self-similarity groups to quadratic roots and continued fractions, thus classifying them up to similarity.

Contribution

It provides necessary and sufficient conditions for similarity relations between packings and explicitly describes their self-similarity groups, connecting geometric packings with algebraic number theory.

Findings

Packings with non-trivial self-similarity relate to roots of quadratic polynomials.

Classification of packings up to similarity using continued fractions.

Explicit description of the self-similarity group for each packing.

Abstract

We consider Apollonian circle packings of a half Euclidean plane. We give necessary and sufficient conditions for two such packings to be related by a Euclidean similarity (that is, by translations, reflections, rotations and dilations) and describe explicitly the group of self-similarities of a given packing. We observe that packings with a non-trivial self-similarity correspond to positive real numbers that are the roots of quadratic polynomials with rational coefficients. This is reflected in a close connection between Apollonian circle packings and continued fractions which allows us to completely classify such packings up to similarity.