There are several GeoGebra applets which demonstrate the Mandelbrot set, that is, the subset of the complex plane of numbers [i]z[/i] such that [math](\ldots((z^2+z)^2+z)^2+\ldots +z)^2+z[/math] is bounded.[br]One of my favorites is [url=https://danpearcymaths.wordpress.com/]Daniel Piercy[/url]'s work "[url=https://tube.geogebra.org/m/40167]Painting the Mandelbrot Set[/url]". The original one is a little bit slow in a browser, so I minimized and modified it to work faster.

Here the magenta points are elements of the Mandelbrot set, black points are outside, blue points are at the boundary.[br]Another approach is the following one which was copied from Mike Borcherds's collection:

Here the black background points are elements of the set, and yellow-green points are outside, other colors mean the boundary. The draggable red point shows the orbit of [i]z[/i] in the formula above.[br]For the best experience perhaps it's even better to check out the [url=http://xaos.sf.net]XaoS[/url] application which very efficiently computes and zooms in the Mandelbrot set. Its zooming engine has also been [url=http://jblang.github.io/XaoSjs/]implemented[/url] in pure JavaScript by J. B. Langston.