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2009-12-18 Jonathan McCabe
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December 7th 2009
The beauty of roots
John Baez reports here on a cool discovery by Dan Christensen. It goes as follow. Considers polynomials in one complex variable z, of degree less than a fixed number n, of the form c_n z^n + c_(n-1) z^(n-1) + ... + c_1 z + c_0. Look at all the polynomials whose coefficients c_i are both integer and comprised between +k and -k, for some integer k. Now plot all the roots of these polynomials in the complex plane. You will get nice pictures with fractal structures, like the one below.
It is especially interesting that some of these fractals look very much like what you can get from affine Iterated function systems (IFS). Even if it's a pretty circonvoluted and inefficient way of drawing IFS, this calls for an explanation... More images, explanations and references on John Baez's page.
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