The sensual Apollonian circle packing
Number Theory
2014-03-13 v3 Metric Geometry
Abstract
The curvatures of the circles in integral Apollonian circle packings, named for Apollonius of Perga (262-190 BC), form an infinite collection of integers whose Diophantine properties have recently seen a surge in interest. Here, we give a new description of Apollonian circle packings built upon the study of the collection of bases of Z[i]^2, inspired by, and intimately related to, the `sensual quadratic form' of Conway.
Cite
@article{arxiv.1208.4836,
title = {The sensual Apollonian circle packing},
author = {Katherine E. Stange},
journal= {arXiv preprint arXiv:1208.4836},
year = {2014}
}
Comments
This version substantially revised: proof vastly streamlined. 33 pages, color figures. Work on quadratic forms related to circles is being moved to a separate paper, but can still be viewed in previous versions of this document on arXiv