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I am trying to continuous coloring the inside point for a fractal image,such as $z \to z^2+C$. For those outside point, we can use the escape iteration to determine the color, just as the description in the Wikipedia. However, how to coloring those inside point, which wouldn't escape? I mean use some good color. Has anyone done it before and can give some suggestions?

Best Regards,

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    $\begingroup$ In the very same article you mention, there is a method described. See the "interior distance estimation". $\endgroup$
    – Raskolnikov
    Commented Dec 22, 2011 at 8:30
  • $\begingroup$ See : en.wikibooks.org/wiki/Fractals HTH $\endgroup$
    – Adam
    Commented Oct 14, 2012 at 19:03

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Typically, if a polynomial Julia set has non-empty interior, then there will be an attractive periodic orbit. Thus, the interior can be colored in much the same way that the exterior is colored - i.e., the color can be based on the number of iterates it takes to get close to the attractive orbit.

This is exactly the procedure that Wolfram Alpha uses to generate "escape time" Julia sets. By default, the Julia set is displayed using an inverse iteration algorithm, but there is a toggle switch that allows you to change to an escape time image, to get an image like so:

enter image description here

While this is not a color image, colors easily could have chosen, rather than shades of gray. One advantage of this approach is that it is equally applicable to rational functions, as well as polynomials. For example:

enter image description here

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