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2009-12-07 The beauty of roots

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A blog about algorithmic art and fractal aesthetic. Click here to subscribe to the RSS feed.

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.
Roots of polynomials of degree less than 5 with integer coefficients

A plot of the roots of all the polynomials of degree less than 5 with integer coefficients comprised between -4 and 4, by Dan Christensen.

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.
Zoom on a plot of the complex roots of polynomials

A zoom around (1/2)exp(i/5) on the plot of all the polynomials of degree 24 with coefficient either 1 or -1, by Sam Derbyshire

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