The sensual Apollonian circle packing
  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. 
Katherine E. Stange
math.NT math.MG