Apollonian circle packings of the half-plane

Details Apollonian circle packings of the half-plane Apollonian circle packings of the half-plane

2011-02-08

arxiv.org/abs/1102.1628v1

Mathematics

Metric Geometry

35 pages

Scientific paper

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.

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