Abstract

The circle is arguably the most studied object in mathematics, yet I am here to tell the tale of circle packing, a topic which is likely to be new to most readers. These packings are configurations of circles satisfying preassigned patterns of tangency, and we will be concerned here with their creation, manipulation, and interpretation. Lest we get off on the wrong foot, I should caution that this is NOT twodimensional “sphere ” packing: rather than being fixed in size, our circles must adjust their radii in tightly choreographed ways if they hope to fit together in a specified pattern. In posing this as a mathematical tale, I am asking the reader for some latitude. From a tale you expect truth without all the details; you know that the storyteller will be playing with the plot and timing; you let pictures carry part of the story. We all hope for deep insights, but perhaps sometimes a simple story with a few new twists is enough—may you enjoy this tale in that spirit. Readers who wish to dig into the details can consult the “Reader’s Guide ” at the end. Once Upon a Time … From wagon wheel to mythical symbol, predating history, perfect form to ancient geometers, companion to π, the circle is perhaps the most celebrated object in mathematics.

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