A generalization of Apollonian packing of circles

Three circles touching one another at distinct points form two curvilineartriangles. Into one of these we can pack three new circles, touching eachother, with each new circle touching two of the original circles. In sucha sextuple of circles there are three pairs of circles, with each of thecircles in a pair touching all four circles in the other two pairs.Repeating the construction in each curvilinear triangle that is formedresults in a generalized Apollonian packing. We can invert the wholepacking in every circle in it, getting a “generalized Apolloniansuper-packing”.Many of the properties of the Descartes configuration and the standardApollonian packing carry over to this case. In particular, thereis an equation of degree 2 connecting the bends (curvatures) of a sextuple;all the bends can be integers; and if theyare, the packing can be placed in the plane so that for each circle withbend $b$ and center $(x,y)$, the quantities $bx/\sqrt 2$ and $by$ areintegers. The construction provides a generalization of the Farey seriesand the associated Ford circles.

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Published 1 January 2010