A generalization of Apollonian packing of circles

Abstract

Three circles touching one another at distinct points form two curvilinear triangles. Into one of these we can pack three new circles, touching each other, with each new circle touching two of the original circles. In such a sextuple of circles there are three pairs of circles, with each of the circles in a pair touching all four circles in the other two pairs. Repeating the construction in each curvilinear triangle that is formed results in a generalized Apollonian packing. We can invert the whole packing in every circle in it, getting a “generalized Apollonian super-packing”. Many of the properties of the Descartes configuration and the standard Apollonian packing carry over to this case. In particular, there is an equation of degree 2 connecting the bends (curvatures) of a sextuple; all the bends can be integers; and if they are, the packing can be placed in the plane so that for each circle with bend b and center (x, y), the quantities bx/ √ 2 and by are integers. The construction provides a generalization of the Farey series and the associated Ford circles.