Mandelbrot set

The Mandelbrot set is a fractal that is defined as the set of points c in the complex number plane for which the iteratively defined sequence

z_n+1 = z_n² + c

with z₀ = 0 does not tend to infinity. If we reformulate this without complex numbers, replacing z_n with the point (x_n, y_n) and c with the point (a, b), then we get

x_n+1 = x_n² - y_n² + a

and

y_n+1 = 2x_ny_n + b.

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. Those points within the Mandelbrot set correspond precisely to the connected Julia sets, and those outside correspond to disconnected ones.

Plotting the set

It can be shown that once the modulus of z_n is larger than 2 (in cartesian form[?], when x_n² + y_n² > 2²) 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. For points inside the Mandelbrot set, i.e. values of c for which z_n doesn't tend to infinity, the calculation never comes to such an end, so it must be terminated after some number of iterations determined by the program. This results in the displayed image being only an approximation to the true set.

Whilst it is of no mathematical importance, most fractal rendering programs display points outside of the Mandelbrot set in different colours depending on the number of iterations before it bailed out, creating concentric shapes, each a better approximation to the Mandelbrot set than the last.

Freeware fractal generators

• Fractint (http://spanky.triumf.ca/www/fractint/fractint.html) (ports available for most platforms)
• "Makin' Magic Fractals (http://skyscraper.fortunecity.com/terabyte/966)"
• ChaosPro (http://www.chaospro.de) - for Microsoft Windows
• Xaos (http://xaos.sourceforge.net/) - Realtime generator - Windows, Mac, Linux, etc.