### Recent Posts

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##### Fractal movie gallery / octofold study #2
« Last post by udo2013 on Yesterday at 02:34:40 PM »
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##### Fractal movie gallery / Fractal Kelp--"1984: Darkness, Evil, Death & Destruction"--3D Fractals & Music
« Last post by Paigan0 on Yesterday at 01:57:47 PM »
Swim through the Fractal Kelp of the Fractal Realms, to the tune of my EDM tune, "1984: Darkness, Evil, Death & Destruction."

Using Krzysztof Marczak's "Trees" KIFS formula settings, and then adding the Gnarl transformation (for the first time), at 90 degrees X and Y rotation, but with the strength and intensity audio synced to the music, along with the cone height and the glow intensity. I'm also using a tiny bit of Turbulent Displacement in post to make things look underwater wavy.

Settings for the entire thing is below! Down in Fractal Rock!

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##### Image Threads / Re: Fourfold
« Last post by pauldelbrot on Yesterday at 12:04:10 PM »
Marigold Mayhem
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« Last post by mrrudewords on Yesterday at 09:19:18 AM »
Thanks, nice image. Happy to make improvements. Feel free to keep the suggestions coming.
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##### Image Threads / Re: Experiments with colouring and layers
« Last post by Sabine62 on Yesterday at 09:14:12 AM »
Nova Pow3
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« Last post by Sabine62 on Yesterday at 09:09:40 AM »
Thank you, mrrudewords!
Julia is fine now, non-integer powers are great to have, and I am Very happy with the option to stop a render
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##### Fractal movie gallery / fractal art
« Last post by udo2013 on Yesterday at 05:47:13 AM »

the geometry of the emptiness and fractal geometry have one thing in common :: both are infinite
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##### Fractal Mathematics And New Theories / Re: Holomorphic Mandelbrot extensions
« Last post by gerrit on Yesterday at 02:15:04 AM »
$$z_0 = \sqrt{-pq/4}\\ w_0 = -w_0\\ z \leftarrow z^2 + qw +c\\ w \leftarrow w^2 + pz +c$$
p = -1, q = 0.95. First full set, then a closer look at all the different scary shapes on the real axis, last one a deeper zoom with weird mins (black).
Distance estimation does not work flawlessly, those pink=purple circles are artifacts. Not sure if there is a better DE than just copying the M-set formula.
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##### Fractal Mathematics And New Theories / Re: Holomorphic Mandelbrot extensions
« Last post by mrrudewords on Yesterday at 12:40:10 AM »
A zoom on that same formula. But I expect this falls in line with formulas you say show m-set shapes but with strange decorations. So not as original as you're looking for?
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##### Fractal Mathematics And New Theories / Re: Holomorphic Mandelbrot extensions
« Last post by mrrudewords on Yesterday at 12:16:31 AM »
While the classical Mandelbrot set $$z \leftarrow z^2 +c$$ remains the king of (2D) fractals, I'm interested in generalizations.
Agree Mandelbrot seems to be unbeatable.

Personally I find all the non-holomorphic extensions like burning ship unappealing.
Also agree. Burning ship is ugly as! There is one place to zoom that gets semi-appealing results but only when I'm drunk!

Recently pauldelbrot posted a formula based on two complex variables, "Mandelbrot foam", which I find extremely interesting in that it is really different from the M-set; no familiar patterns appear at all, yet it is not "ugly" like the non-holomorphic formulas (IMHO of course).
Yes, mandelfoam has some strangely organic results but less of the obviously deformed Mandelbrot shapes you see on those.

I'm interested in strategies to find such interesting fractals, but don't know how to approach it. Making tweaks to M-foam gave some interesting results, but how take a more mathematically informed approach?

For example, what is a "critical point" when you have multiple variables? Is it where the Jacobian is zero? Or just the determinant (local magnification)?

More generally, how do you come up with "interesting" formulas?

I realize my questions may not have a real (or complex answer but any ideas would be appreciated.
I'm totally lost!

My approach is very much just coding. I can take a formula and tweak stuff around, add things in and basically experiment, but I have no idea how the mathematics are changed by my whimsical actions. In-fact I am lost looking through many of your's and pauldelbrot's posts where you speak of certain mathematical theories. I really like to see the beauty of the results and take pleasure in the fact that the images have been produced using these formulas and I'd love to understand them better.

Recently I've been messing with combining formulas in the hopes of something interesting. Most of the time, as you'd probably expect, this results in the hideously deformed offspring of the two.

But Nova/Newton seems to result in some nice images.
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