### Looking for Mandelbrot Butterfly explorers and developers

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#### JonathanD

• Fractal Fanatic
• Posts: 36
• Fractal Physics Guru

#### Re: Looking for Mandelbrot Butterfly explorers and developers

« Reply #15 on: February 02, 2020, 05:33:19 AM »
Note:

Setting the iterand to the value: bailout + 1 means it will kick out of the loop and assign a color, as though the butterfly wing or disc was one of the repeller sets (the color bands lining the edges of the set).  Obviously; if these color bands are not being displayed that method will not work, and the butterfly wings will not be colored in either.  However; it is possible to suppress the repeller sets lining the edge and display the butterfly parts, with a slight adjustment.

More later,

Jonathan
Jonathan J. Dickau

• 3f
• Posts: 1979

#### Re: Looking for Mandelbrot Butterfly explorers and developers

« Reply #16 on: February 02, 2020, 05:40:27 AM »
Well what I tried was to replace $$z$$ by $$z+\psi$$ with $$\psi$$ as Grassman variable (i.e., $$\psi^2=0$$, but $$\psi \neq 0$$).

So we get the super iteration:
$$z+\psi \leftarrow z^2+2z\psi + c + \gamma$$
with $$\gamma$$ the Grassmann component of the pixel.

Unfortunately this just amounts to
$$z \leftarrow z^2 +c\\ \psi \leftarrow 2z\psi + \gamma$$

so no non-trivial mixing of the Grassman variable and $$z$$.

I don't quite get the relevance of G2 Lie algebra, perhaps the graded Lie-algebra's used in supersymmetry speculations in theoretical physics can be used?

#### JonathanD

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• Posts: 36
• Fractal Physics Guru

#### Re: Looking for Mandelbrot Butterfly explorers and developers

« Reply #17 on: February 02, 2020, 07:40:32 AM »
You have cool ideas Gerrit...

And that substitution does sound interesting.  I got a package for NodeJS from Rick Lockyer for Octonion Maths, that I need to take more time to play with.  But now it's about time to catch some sleep.

More when there is time,

Jonathan

#### JonathanD

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• Posts: 36
• Fractal Physics Guru

#### Re: Looking for Mandelbrot Butterfly explorers and developers

« Reply #18 on: February 04, 2020, 03:35:53 AM »
Look for my formula files...

Now in the Fractal software Downloads section.  Also copied below.  ZIP has instructions and CFM file for Chaos Pro.

Best,

JJD

#### C0ryMcG

• Strange Attractor
• Posts: 92

#### Re: Looking for Mandelbrot Butterfly explorers and developers

« Reply #19 on: February 10, 2020, 07:38:28 AM »
I have a little Javascript project that I'm intending to make as flexible as possible, and I used that to implement the method you described as a coloring model, rendering Only the butterfly. It resulted in some pretty interesting results, I think. I don't know if it'll help you with your ultimate goal of linking the Mandelbrot set to particle physics theory, but at least it looks nice!

I can share a code snippet if it's desired... but basically it looks at a list of points, and for the first set of points that declines in radius from center, it performs this:

colorIndex = d1 * Math.max(1/(d3 - d2),1/(d2 - d1));

Then it selects a color from a looping array (with some lerping to get nice in-betweens), and sends it to the screen.
Any thoughts?

#### gerson

• Fractal Furball
• Posts: 292

#### Re: Looking for Mandelbrot Butterfly explorers and developers

« Reply #20 on: February 10, 2020, 02:39:20 PM »
Thanks for sharing the formulas. Liked very much.

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