Learning Your Way Around

The Branching Rule

As you venture deeper into the Mandelbrot Set, you will often encounter a progression illustrated below, in which the symmetry doubles from two arms to four arms to 8 arms to 16, etc... The technical name for this behavior is Bifurcation, and it is one of the most fundamental elements of fractals and chaos.



A sequence of zooms into the spike at the far left of the Mandelbrot Set, showing 2 arms bifurcating into 4, which then bifurcate into 8 arms, etc. Click mouse to Zoom In <Ctrl>-Click mouse to Zoom Out Add <Shift> to Zoom faster

This Bifurcation or doubling rule is universal, so it applies everywhere in the Mandelbrot Set.
<Ctrl>-click the image below to zoom out and see the sequence of bifurcations in reverse from a different location.

Did you find where in the Mandelbrot Set that detail came from? Have you observed that at the bottom of the sequence of doubling there is always a little replica of the entire Mandelbrot Set? This is an amazing discovery! In the next chapter we will explore these little copies of the Mandelbrot Set in more detail.

But first we will learn about the Periodicities of the Mandelbrot Set.