The numbers represent the curvatures (1/radius) of each circle, with the boundary circle having curvature 1 and the two largest circles having curvature 2 (radius 1/2).
At SIAM News
we’re always happy (but not particularly surprised) to have outside confirmation of the talents of our writers. A case in point is the awarding of the MAA’s Chauvenet Prize to Dana Mackenzie at the Joint Meetings in San Antonio in January.
The illustration shown here is from “A Tisket, a Tasket, an Apollonian Gasket” (American Scientist, Vol. 98, January–February 2010), the paper for which Mackenzie was honored. Here’s his summary of the paper: “Start with three mutually tangent circles, insert a fourth circle tangent to all three, and iterate. If the four circles have curvatures that are integers, then all subsequent curvatures are integers too, and the construction yields an integral Apollonian circle packing.”
An earlier (2012) recipient of the Joint Policy Board for Mathematics Communication Award, Mackenzie nonetheless points out that gratification for writers can be long delayed or nonexistent. For SIAM News he often chooses the topics he’d like to write about; what usually captures his attention is some new mathematical approach to a physical problem. A few recent examples of his articles are: “Synthetic Biology, Real Mathematics” (December 2014), “Kadison–Singer Problem Solved” (January/February 2014), and “Smells Like a Traffic Jam” (November 2013).
A compliment to a writer is never amiss. Suggested topics matched to the interests of our writers are always welcome too—at firstname.lastname@example.org.