Thirteen years ago, I wrote an article for American Scientist called "A Tisket, a Tasket, an Apollonian Gasket" (January-February 2010), which explored the fascinating mathematics and history behind a certain type of fractal foam-the "Apollonian gaskets" of the title. Recently, I've begun to see that history itself is like a fractal: When you take even the smallest event and magnify it, it springs to life and turns out to have a whole story of its own. One sentence in particular from my article nagged at me: "In 1643, in a letter to Princess Elizabeth of Bohemia, the French philosopher and mathematician Rene Descartes correctly stated (but incorrectly proved) a beautiful formula concerning the radii of four mutually touching circles." This offhand comment is not really wrong but also not really right, and what's worse is that I had drawn the idea from articles by other scholars without looking into it critically. In fact, this one sentence raises all sorts of questions. Why was Descartes writing to a princess, and how did they get on the subject of circles? Why did he write to her about this problem, and not to the many other mathematicians he corresponded with? And who was Princess Elizabeth, anyway?
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