Mathematical
Definition - The Mandelbrot
set is defined as the set of points c
in the complex plane for which the iteratively defined sequence z_{n}_{+1}
= z_{n}^{2}
+ c
with z_{0} =
0 does not tend to
infinity.
Approximation
- It can be shown that once the modulus of z_{n} is
larger than 2, then the sequence will tend to infinity, and c
is therefore outside the Mandelbrot set. This value, known as the
bail-out value, allows the calculation to be terminated for points
outside the Mandelbrot set. If the modulus of z_{n}
remains less than 2 after a given number of iterations, then there
is frequently no way to know whether it would diverge if additional
iterations were performed. No matter how many iterations are
performed, any visualization of the Mandelbrot set will inevitably
include some points that would have diverged if still more
iterations were performed. Therefore, all visualizations of the
Mandelbrot set are approximations that overestimate the size of the
set. As the number of maximum iterations is increased, the image of
the Mandelbrot set gradually shrinks toward a more accurate shape.
Color
- Mathematically speaking, the picture of the Mandlebrot set is
black and white. Either a point is in the set (usually colored
black) or it is not (colored white). Most fractal rendering
programs display points outside of the Mandelbrot set in different
colors depending on the number of iterations before it bailed
out. This is what creates the multi-colored images. Source for the above isen.wikipedia.org/wiki/Mandelbrot_set.
Fractalus - www.fractalus.com
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