Coloring the Mandelbrot Set with Hues

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This image was created by taking the standard Mandelbrot set algorithm and coloring each pixel according to the argument of the last iterate. This is a simple demonstration of how using information from the last iterate can give you more information for coloring a pixel than just using the iteration count.

An interesting feature of this image is that you can see the period doubling of the argument through each iteration. (Look in the image for where the band of red repeats in the level sets.) Why does the phase double every iteration? The iteration of each point proceeds by squaring the point and adding a constant. When you square a complex number, you double its argument. This is why each iteration doubles the phase of the iterate.

The image was computed by software that I wrote and rendered with 3x oversampling and then downsampled to its final size with a Mitchell filter.