Greg M. Johnson <gre### [at] my-dejanewscom> wrote:
: No, the iteration amount of 30 means your trees are demarcated into 30
: different possible heights.
: It does nothing for the complexity of structure in x-z plane.

  Sorry, but I have to disagree. And this is easy to prove:

  Calculate a regular mandel with 1 iteration. You get a black (or
interior-colored) circular shape and one color in the exterior (the shape
is exactly circular because every point that is inside your threshold
radius will be considered "inside" and all the other points will be
"outside").

  Calculate the mandel with 2 iterations.
  Now you get a band of different color (supposing you are using a color
map that allows you to distinguish well between two adjacent colors)
inside the abovementioned circle and a black shape which is less
circular, but still with a smooth edge.

  Calculate the mandel with 3 iterations.
  Another color band will appear inside the previous one and the black
"inside" area will be smaller and less circular-shaped, but still quite
smooth.

  In no way can you say that that black area has infinite complexity in
its border. It's quite smooth.

  With more iterations (tens, hundreds...) the black "inside" area will
"twist" more and more and its border will be less smooth with each increased
iteration amount, but if you zoom it large enough, you will always find
that the border is smooth, without infinite complexity.

  To get the true mandelbrot fractal, with infinitely complex border
(and infinitely long), you'll have to make an infinite number of
iterations. This is also its mathematical definition.
  Since we can't calculate infinite iterations we have to calculate a
finite number of them. Fortunately, since the resolution of our screens
is limited, after going up to certain iterations amount, the lack of
complexity will be buried in the pixel resolution of the image and thus
we will not notice the difference.
  That's why a finite number of iterations is enough to get a good-enough
approximation.
  But it also means that if we zoom the fractal enough, we have to increase
the iterations amount.

-- 
main(i,_){for(_?--i,main(i+2,"FhhQHFIJD|FQTITFN]zRFHhhTBFHhhTBFysdB"[i]
):_;i&&_>1;printf("%s",_-70?_&1?"[]":" ":(_=0,"\n")),_/=2);} /*- Warp -*/

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