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 noncompact 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 modifications of the problem are investigated, and relations to other problems, such as the universal scaling of circle maps, are pointed out.

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