Consider the mapping . The Mandelbrot set consists of those complex numbers such that the iterates of do not tend to infinity as . Points with an iterate greater than 2 in absolute value diverge.
As you drag the blue point or use "run" to get a random path, the blue line joins 's first 20 iterates under . also leaves a track of green or red points according to whether is inside the core region of the Mandelbrot set or not.
Iterates that leave the gray disk of radius 2 can be excluded as possible candidates for the Mandelbrot set because their next iteration will again be outside that critical disk.