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# Journal of Combinatorics

## Volume 3 (2012)

### Number 1

### Apollonian circle packings of the half-plane

Pages: 1 – 48

DOI: http://dx.doi.org/10.4310/JOC.2012.v3.n1.a1

#### Authors

#### 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.

#### Keywords

Apollonian circle packings, similarity, continued fractions

#### 2010 Mathematics Subject Classification

11A55, 52C26