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.
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