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Growing Apollonian Packings
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Colin Mallows

Avaya Labs

Basking Ridge, NJ 07920

USA

**Abstract:**

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
three-dimensional case is difficult. There is an ambiguity in how
the successive generations are defined. We are unable to give general
results for this case.

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Received September 26 2008;
revised version received January 2 2009.
Published in *Journal of Integer Sequences*, January 3 2009.

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