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Mandelbrot Set found as a rule for WolframModel Hypergraphs

Posted 1 month ago
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Hi all!

I'm new around here, so I apologise if my post is in the wrong place, or is irrelevant, or perhaps ignorant. I have been having a blast playing around with Wolfram Models, and today I stumbled upon an interesting set of evolutions: Mandelbrot Set Emergence

I wonder if anybody else has found any results like this previously. I was hoping somebody with a bit more experience could do some extra analysis on this as it does appear to exhibit some strange behaviour, for example, the entire hypergraph flips on the 8th evolution:

8th Evolution Flip

This seems very very strange to me, as I had previously believed that the mandelbrot set could only be generated in the complex plane? This seems to suggest that a simple arbitrary rule involving simple relationships can generate something previously only found in the complex plane. Please let me know if I am way off in this, as I do not pretend to be a high-level mathematician, I am in the infancy stages of this pursuit.

Some interesting discussion and other similar hypergraphs are welcomed! I really would love to see more iterations of this rule however I currently only have access to the Cloud version of the Wolfram environment, so I am limited to 9 evolutions. Looking forward to a cool chat!

5 Replies
Posted 1 month ago

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.

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 enter image description here obtained from the rule {{x, y}} → {{z, z}, {x, z}, {y, z}} enter image description here

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

Anton

Posted 1 month ago

Hey Anton,

Thanks for the informative comment! I'll be sure to read more of the technical document as I had a hunch that this kind of pattern would very easily and naturally emerge from a repeated set replacement algorithm. But I was still surprised when I initially saw the similarities!

Warm regards, Jacob

Wow! Didn't even know that was possible..! Impresssive.

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..

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