Apollonian tiling, the Lorentz group, and regular trees.

The Apollonian tiling of the plane into circles is analyzed with respect to its group properties. The relevant group, which is non-compact and discrete, is found to be identical to the symmetry group of a particular geometric tree-graph in hyperbolic three-space. A linear recursive method to compute the radii is obtained. Certain modiications of the problem are investigated, and relations to other problems, such as the universal scaling of circle-maps, are pointed out.