Growing Apollonian Packings

In two dimensions, start with three mutually tangent circles, with disjoint interiors (a circle with negative radius has the point at infinity in its interior). We can draw two new circles that touch these three, and then six more in the gaps that are formed, and so on. This procedure generates an (infinite) Apollonian packing of circles. We show that the sum of the bends (curvatures) of the circles that appear in successive generations is an integer multiple of the sum of the bends of the original three circles. The same is true if we start with four mutually tangent circles (a Descartes configuration) instead of three. Also the integrality holds in three (resp., five) dimensions, if we start with four or five (resp. six or seven) mutually tangent spheres. (In four and higher dimensions the spheres in successive generations are not disjoint.) The analysis in the threedimensional case is difficult. There is an ambiguity in how the successive generations are defined. We are unable to give general results for this case. 1 Reflection and the Apollonian group We draw on the definitions and analysis in [1, 2]. The “bend” of a circle or sphere is its curvature, = 1/radius. In n dimensions, a “Descartes configuration” consists of n + 2 mutually tangent spheres with disjoint interiors. The bends bi of these spheres satisfy the Soddy-Gosset relation (see [2]) n( ∑