Mandelbrot Foam

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Offline Softology

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« on: November 27, 2018, 02:39:43 AM »
Is there a post/topic explaining the "Mandelbrot Foam"?

This post
https://fractalforums.org/fractal-mathematics-and-new-theories/28/mandelbrot-foam-math/1665/msg8427#msg8427
shows the formulas/iterations as
w=qw/z
z=z^2+w^2+c

What should q and w be initialized to?  What are some interesting variations of q and w?

Trying random values is only getting me blank images.

Offline gerrit

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« Reply #1 on: November 27, 2018, 03:05:17 AM »
That's covered in the replies.

Offline Softology

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« Reply #2 on: November 27, 2018, 08:52:34 PM »
With this code

Code: [Select]
q.x=-0.5;
q.y=0.0;
w.x=0.2;
w.y=0.0;
z = dvec2(1.0,0.0);
for(i=0; i<maxiters; i++)
{
w=cdiv(cmul(q,w),z);
z=cadd(cadd(csqr(z),csqr(w)),c);
//magnitude bailout check here
}

I got this result


Does that look right to you?

What are the q, w and z for the 2 images in this post?
https://fractalforums.org/fractal-mathematics-and-new-theories/28/mandelbrot-foam-math/1665/msg8502#msg8502

Thanks,
Jason.

Offline gerrit

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« Reply #3 on: November 28, 2018, 12:49:21 AM »
As the post says: q=-1/2. Any w, z with w^2+z^2=0 is a critical point, looking best imho, but if course you don't have to start there.

Offline Softology

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« Reply #4 on: November 28, 2018, 08:48:13 PM »
Any w, z with w^2+z^2=0 is a critical point

This is the part I don't understand.  Changing w does give me unique results.  What does "w^2+z^2=0 is a critical point" mean?  What specific w and z values did you use for those 2 example images (so I can replicate and make sure my code is OK)?

Thanks,
Jason.

Offline gerrit

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« Reply #5 on: November 29, 2018, 01:35:59 AM »
Initial \( w=iz \) for any z (except 0) gives same result.
Second image is probably bug, as stated in post, so no idea what it is.
Critical point: use search engine.

Offline 3DickUlus

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« Reply #6 on: November 29, 2018, 01:54:18 AM »
"any point where the function's derivative is zero"
Resistance is fertile... you will be illuminated!

https://en.wikibooks.org/wiki/Fractals/fragmentarium

Offline kosalos

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« Reply #7 on: December 01, 2018, 01:24:43 AM »
Softology's algorithm works well for me.
Only change was to allow run-time edits of the z,q and w initial values.

Offline Softology

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« Reply #8 on: December 02, 2018, 07:56:25 AM »
Here are a few 4K zooms into various Mandelbrot Foam fractals.


Offline gerrit

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« Reply #9 on: December 02, 2018, 09:04:20 AM »
Here are a few 4K zooms into various Mandelbrot Foam fractals.
Nice.


clip
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