Journal of Combinatorics

Volume 3 (2012)

Number 1

Apollonian circle packings of the half-plane

Pages: 1 – 48



Michael Ching (Department of Mathematics, Amherst College, Amherst, Mass., U.S.A.)

John R. Doyle (Department of Mathematics, University of Georgia, Athens, Ga., U.S.A.)


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

11A55, 52C26

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