A generalization of Apollonian packing of circles

A generalization of Apollonian packing of circles Gerhard Guettler Colin Mallowsy 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 con guration 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=p2 and by are integers. Recently there has been renewed interest in a very old idea, that of Apollonian packing of circles, in which an initial con guration of three mutually tangent circles is augmented by repeatedly drawing new circles in each curvilinear gap. See for example Mumford et al [8]. We can also study \super-Apollonian" packings which are obtained by repeatedly inverting an Apollonian packing in every circle in it. It is a remarkable fact that Apollonian and super-Apollonian packings exist in which all the bends (curvatures) are integers. This property was studied in detail by Graham et al [3], and the group theory associated with these packings has been studied by the same authors [4-6]. Also, if all the bends are integers, the super-Apollonian packing can be placed in the plane so that all the \bend times center" quantities are integers. Several extensions of the Apollonian idea have been studied, for example Mauldon [7] studied con gurations in which adjacent circles do not touch but have constant \separation". Our own interest lies in extending these ideas in new directions, particularly by packing not one but three circles within each triangular gap, thus forming sextuUniversity of Applied Sciences Giessen Friedberg (Germany), [email protected] yAvaya Labs, Basking Ridge NJ USA 07920, [email protected] ples of circles, and in exploring the degree to which the theory associated with the classical packings can be extended to cover this case. We nd that all the bends in such a generalized packing can be integers; and there are results relating to the positions of the centers of the circles that directly generalize those found by Lagarias et al [2] in the classical Descartes-Apollonian case. Figure 1 shows the four possible con gurations of a sextuple. There can be zero, one, or two circles wth bend zero (i.e. straight lines), and at most one bend can be negative, as in case (a). (a) (b) (c) (d)