Journal of Combinatorics
Volume 3 (2012)
Apollonian circle packings of the half-plane
Pages: 1 – 48
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.
Apollonian circle packings, similarity, continued fractions
2010 Mathematics Subject Classification