Mandelbrot set

The Mandelbrot set is a fractal that is defined as the set of points in the complex number plane that do not tend to infinity when iterating

z_n+1 = z_n² + c

where z_i and c are complex numbers. If we reformulate this without complex numbers and assume that z = (x, y) and c = (a, b) then we get

x_n+1 = x_n² - y_n² + a, and

y_n+1 = 2x_ny_n + b.

It can be shown that once the modulus of z is larger than 2 (in [cartesian form]?, when x²+y²>2²) the point c will tend to infinity. This value, known as the Bailout value enables a large number of iterations to be skipped when calculating points outside the Mandelbrot set. Whilst it is of no mathematical importance, most fractal rendering programs display points outside of the Mandelbrot set depending on the number of iterations before it bailed out, creating concentric shapes, each a better approximation to the Mandelbrot set.

The Mandelbrot set was created by Benoit B. Mandelbrot as an index to the Julia? Sets: each point in the complex plane corresponds to a different Julia set, and those points within the Mandelbrot set correspond precisely to the connected Julia sets.

-

Freeware Fractal Generators:

http://spanky.triumf.ca/www/fractint/fractint.html : Fractint (ports available for most platforms)
http://skyscraper.fortunecity.com/terabyte/966 : Makin' Magic Fractals
http://www.chaospro.de : ChaosPro?