Mandelbrot

The Mandelbrot set is the most well-known fractal type. Although it is calculated by a simple formula, it is incredibly complex. As you zoom in, more and more ever-changing detail becomes visible, such as little "baby" Mandelbrot sets and all kinds of spirals.

Because the Mandelbrot set lends itself well to basic zooming and exploring, it is a good starting point if you are new to fractals. It is available as a fractal formula in Standard.ufm and as a fractal formula plug-in in Standard.ulb.

The formula provides the following parameters:

Starting point

For the standard Mandelbrot set, this should be set to (0, 0). Other values create distorted shapes that can be interesting, but they are usually not as well-formed as the standard set. Try (0, -0.6), for example.

Power

Specifies the exponent. The default value is (2, 0), resulting in the classic equation.

z = z² + c

Try (3, 0) and (4, 0) and so on to increase the number of main "buds". Non-integer values for the real part of the exponent will interpolate between these well-formed sets. If the imaginary part is not zero, the fractal will be further distorted.

Bailout value

Specifies the magnitude of z that will cause the formula to stop iterating. To obtain the "true" Mandelbrot set, this should be set to 4 or larger. Larger values tend to smooth the outside areas.

With the Basic coloring algorithm and the Color Density set to 4, try the bail-out values 4 and then 16 to see the difference.

Some coloring algorithms require specific bail-out values for good results.

Notes

The Mandelbrot set is also available as a more efficient built-in formula with fewer options. See Mandelbrot (Built-in).
The Mandelbrot set also acts as a map of Julia sets. Use Switch mode to switch to related Julia sets.