The Mandelbrot set is defined by the recurrence equation of the form \(z_{n+1} = f(z_n, C)\), where \(C\) is come complex number, and \(z_0 = C\). All points for which the recursion does not escape to infinity are within the set. Here all the black points are within the set, since they do not escape to infinity within a certain number of steps. Infinity here is defined simply by a certain radius around the origin. All points for which the equation does escape, are colored accoring to how quickly the equation escapes.

By clicking on the graph, you can see the orbit for that point. This allows a quick glance at where the convergence is for a given point, and a lot of interesting patterns emerge.

Note: Some of the graphs might take a while to render, as the equation has to be calculated for each pixel individually!