We consider the distribution function N(x) of the curvatures of the disks in the Apollonian packing of a curvilinear triangle. That is, N(x) counts the number of disks in the packing whose curvatures do not exceed x. We show that log N(x)/log x approaches the limit S as x tends to infinity, where 5 is the exponent of the packing. A numerical fit of a curve of the form y = Ans to the values of A...

Bounded Apollonian circle packings (ACP’s) are constructed by repeatedly inscribing circles into the triangular interstices in a configuration of four mutually tangent circles, one of which is internally tangent to the other three. An integer ACP is one in which all of the circles have integer curvature. In this paper, we give a lower bound on κ(P,X), the number of integers less than X occurrin...

In this paper we introduce a model of spatial network growth in which nodes are placed at randomly selected locations on a unit square in R, forming new connections to old nodes subject to the constraint that edges do not cross. The resulting network has a power law degree distribution, high clustering and the small world property. We argue that these characteristics are a consequence of the tw...

Circle packings are a particularly elegant and simple way to construct quite complicated and elaborate sets in the plane. One systematically constructs a countable family of tangent circles whose radii tend to zero. Although there are many problems in understanding all of the individual values of their radii, there is a particularly simple asymptotic formula for the radii of the circles, origin...