Apollonian circle packings of the half-plane
Metric Geometry
2015-03-18 v2 Number Theory
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
Cite
@article{arxiv.1102.1628,
title = {Apollonian circle packings of the half-plane},
author = {Michael Ching and John R. Doyle},
journal= {arXiv preprint arXiv:1102.1628},
year = {2015}
}
Comments
35 pages; final version -- minor improvements to the exposition from the first version