The formula used by the Complex Number Fractal Generator can be
thought of as a function `f(z, c)`. It is used to
transform `z`, like so:

z_{n + 1} = f(z_{n}, c)

and is iterated until the modulus (absolute value or size) of
`z` is greater than 2. The point is then coloured
according to how many iterations it took to escape. This part of
the routine is common to both Mandelbrot Sets and Julia Sets. The
diference is in the definitions of `c` and `z`.
In a Mandelbrot Set, the initial value of `z` is 0, and
the initial value of `c` is the point on the complex
plane, while in a julia set, `z`'s initial value is the
point on the complex plane and `c` is some constant.
This means that there is only one Mandelbrot Set for a formula,
while there is an infinite number of Julia Sets. In fact, each
value of `c` for a Julia Set, corresponds to a point on
the Mandelbrot Set. Points that are inside the Mandelbrot Set
(the black bit) will give connected Julia Sets, while points
outside will give disconnected sets, and points on the edge (eg. 0
+ 1i) will give Julia Sets with no area.

Some formulas for implemented fractals are:

Mandel: | f(z, c) = z ^{2} + c |

Mandel3: | f(z, c) = z ^{3} + c |

Mandel4: | f(z, c) = z ^{4} + c |

Copyright © 2005 — James Henstridge <james@jamesh.id.au>