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"Circle Packing and Spontaneous Geometry" by Ken Stephenson

The topic of "circle packing" is about configurations of circles having specified patterns of tangency. You can specify how many circles you want and which of them are tangent to which others, and then --- quite amazingly --- find radii and centers for the circles so that they fit together in exactly the desired pattern! Note that the pattern is combinatorial information, specified via a graph. The circles do an involved dance, adjusting radii until each can fit with its prescribed neighbors, and when they have finished, their layout gives us geometric information. This is "spontaneous" in that each circle worries only about itself and its immediate neighbors, yet the resulting configuration has rigid global consequences. The connection between combinatorics and packing geometry is what we will study. Fortunately, the software CirclePack does the work of computing and displaying the results, providing an open experimental platform for investigating this fascinating interplay between combinatorics and geometry.