The sensual Apollonian circle packing

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.

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