Complex Set

Mathematically defined region of the
complex plane...

For the purposes of this discussion I will define a set as a collection of numbers that all follow a particular mathematical rule. A complex set is then a set of complex numbers that all meet the specified criteria.

For example we specify the Mandelbrot set, named for the mathematician Benoit Mandelbrot, to be the set of complex numbers c, for which the expression z=z^{2} + c does not diverge (grow larger without limit) under iteration.
Every number in the complex plane is either in the set or out of it. It turns out that for many mathematical expressions, including the Mandelbrot example, the set boundary in the complex plane is irregular in the extreme. In fact the length of the boundary may be infinite even though the entire set might be enclosed in a circle of rather small radius. We call figures of infinite length but finite extent
fractals.