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extropalopakettle
Posted: Sun Jan 06, 2013 9:59 pm    Post subject: 1

This spins off from the "Fractal A Day" / "Fractals Revived" topic. I thought I'd start this in a separate place so as not to get lost in all the pretty pictures. The intent is to discuss what's behind the pretty pictures. For starters, no harm in linking to wikipedia:

http://en.wikipedia.org/wiki/Mandelbrot_set

http://en.wikipedia.org/wiki/Julia_set

The Mandelbrot Set: All complex numbers C for which the sequence defined by the following remains bounded:

1) Z 0 = 0
2) Z n+1 = Z n 2 + C

There's a theorem that states if the Z value ever goes beyond a distance of 2 from the origin of the complex number plane, it is unbounded - i.e. the Z values will grow and grow. But for some initial C values, the Z values will eventually converge toward a point .... or toward a finite sequence of points.

So programmatically, from the above, we can do the iteration of the above, and see when a C value is not in the Mandelbrot Set. We know it when a Z value goes beyond distance 2 from the origin. We can't quite know that it never does. In practice, we can set some limit on the number of iterations, and if i doesn't go beyond 2 for say, a thousand iterations, it never will.

If we color black the points C on the complex plane for which the sequence remains bounded, the Mandelbrot Set, we get this shape:

Now, the points C outside the set (white above) ... some will cause the sequence to take longer than others to escape beyond the circle of radius 2. We can color them based on this "escape velocity", resulting in something like this:

There are various tricks to make it a little prettier without changing the actual structure, like smoothing out the bands around the set:

And of course, instead of looking at the area bounded by -2 and 2 real, -2i and 2i imaginary, we can focus on a smaller area, like for instance the crack between the main black body above and the largest disk shaped body (sometimes called "Seahorse Valley"):

Zooming in on the structures on the left and right sides of the valley, we see these:

And then in each of those, we see smaller distorted versions of the original shape, "mini-mandelbrots":

All of the above from simple iteration of Z n+1 = Z n 2 + C, where Z starts as 0, and C corresponds to pixels in the image mapped to the complex number plane.

Now, suppose instead we let C start as 0, and let Z correspond to points on the plane, and plotted that. Turns out that would result in a black disk with circular bands around it - not all that interesting. But we can choose a different fixed value for C. These are the Julia Sets. For any complex number C (called the "seed"), its Julia Set is all complex values for Z 0 such that the following sequence remains bounded:

Z n+1 = Z n 2 + C

While zooming in on the details of the edge of the Mandelbrot Set reveals different structures in different areas, each Julia Set has similar structure everywhere. Also, for any C value chosen from near the edge of the Mandelbrot Set, the Julia Set for that C value has structure similar to what's around the Mandelbrot Set at that location. The Mandelbrot Set has been likened to a catalog of all the Julia Sets (next post).