This stunning picture looks extremely complicated (and in fact, it is, it’s a fractal) but the rule to generate it is extremely simple - even more than Mandelbrot’s iterated function. This is my favorite example of how easily a really simple concept can give rise to apparent chaos.
Generate all the polynomials up to a given degree and with coefficients limited to either -1 or 1. Find their (complex) roots, and plot them in the complex plane. Ta-da: you get this!
You can see a “belt” around the unit circle, with some white spaces indicating that our polynomials apparently tend to avoid roots of unity, unless they have exactly those roots. Also notice the dragon fractal-like and feather-like patterns on the inner and outer sides of the belt!
Mathematica ran a couple of hours through this one. I used different colors for polynomial roots of different degrees: the lighter the gray, the higher the power of the polynomial (up to 15). Of course, because of the symmetry, one should only calculate one quadrant of the image. Find more here.

This stunning picture looks extremely complicated (and in fact, it is, it’s a fractal) but the rule to generate it is extremely simple - even more than Mandelbrot’s iterated function. This is my favorite example of how easily a really simple concept can give rise to apparent chaos.

Generate all the polynomials up to a given degree and with coefficients limited to either -1 or 1. Find their (complex) roots, and plot them in the complex plane. Ta-da: you get this!

You can see a “belt” around the unit circle, with some white spaces indicating that our polynomials apparently tend to avoid roots of unity, unless they have exactly those roots. Also notice the dragon fractal-like and feather-like patterns on the inner and outer sides of the belt!

Mathematica ran a couple of hours through this one. I used different colors for polynomial roots of different degrees: the lighter the gray, the higher the power of the polynomial (up to 15). Of course, because of the symmetry, one should only calculate one quadrant of the image. Find more here.