This rule was actually illustrated in Stephen Wolfram's article, "A Class of Models with the Potential to Represent Fundamental Physics" at the beginning of Section 3.5: "Rules Depending on a Single Binary Relation". Stephen however, expressed it as: "{{x,y}}->{{z,z},{x,z},{y,z}}", and surprisingly, did not mention Mandelbrot. I only mention it now because at the time, I too saw the correspondence..

Hi Jacob,

In Stephen's technical introduction paper (https://www.wolframphysics.org/technical-introduction/typical-behaviors/rules-depending-on-a-single-binary-relation/) he specifically mentions this behavior:

Several distinct classes of behavior are visible. Beyond simple lines, loops, trees and radial “bursts”, there are nested (“cactus-like”) graphs such as obtained from the rule {{x, y}} → {{z, z}, {x, z}, {y, z}}

I ran the rule to 10 steps and it does appear to flip after step 8 back and forth.

Anton

Jacob:

I binged some YouTubes on Mandelbrot sets and Mandelbrot-like sets when I was "guaranteed" a few months ago.

If you search for Mandelbrot videos, you will find a few more examples like the one you found.