Greg M. Johnson <gre### [at] my-dejanewscom> wrote:
: As I understand it, it determines the number of possible values (colors)
: returned by the pattern, or if one had an isosurface, the number of possible
: heights to the pattern.

  Well, not really. That's only a side-effect which is used to color the
exterior (and sometimes the interior) of the set.

  The mathematical definition of the Mandelbrot set defines that when the
function is iterated an infinite number of times, if the value of the
function converges to infinity, then the starting complex number does
not belong to the set. If the value doesn't converge to infinity, then
it belongs to the set.
  The mathematical definition of the Mandelbrot set doesn't say anything
about a certain number of iterations. I don't even think that it makes
sense to talk about "iterations". It's just a recursive function, that's all.

  Since in computers we can't iterate an infinite number of times and we
can't compare numbers to infinity, we have to use an approximation.
  Instead of iterating an infinite number of times we iterate a finite number
of times. Instead of comparing to infinity, we compare to a finite value.
  Fortunately a really good approximation can be achieved with rather small
finite values.
  It's easy to prove that if the orbit of the function ever goes outside
a sphere of radius 2, the function will converge to infinity. That's why
the value 2 is used.
  As for iterations, depending on the resolution of the image a rather
small number of them is enough to get a correct result at pixel-level
(that is, the error produced by the size of the pixel is larger than the
error produced by the limited iterations amount). Zooming the image just
increases the resolution and thus the number of iterations has to be
increased accordingly.

  The mandelbrot set is just a set of complex values.
  The colors seen around the set when it's drawn on screen are not part of
the set. They are (usually) generated by taking the number of iterations
before bailout (ie. before the orbit went outside the circle of radius 2)
and mapping this value to a color.
  That's why the number of iterations affects the colors used to draw the
set.
  It wouldn't be true to say that the colors have nothing to do with the
set itself, since they have. There's a property in the function that makes
points closer to the border of the set take longer to bailout. Because of
this, points closer to the set will take more iterations to bailout and
thus it will affect their color. This can bring out the shape of the set
although the set itself is not visible in some parts (because it's so small).
  However, the coloring according to the number of iterations is purely
artificial and has little to do with the Mandelbrot function itself (except
for the property described above).

-- 
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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