• January 28, 2022, 01:01:10 PM

Author Topic:  Very Slight Variations on the Mandelbrot Exponent  (Read 289 times)

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Offline BrokenParameter

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Very Slight Variations on the Mandelbrot Exponent
« on: January 12, 2022, 05:06:20 AM »
Hello fellow fractal enthusiasts!

Lately I've been exploring slight variations to the Mandelbrot function, in particular tiny increases to the exponent (e.g. z^2.00001).  There are some interesting patterns to be found by zooming into the antenna area, often beyond -1.4, sometimes even geometric patterns that I didn't expect at all.

I've attached some examples.  I've been using ChaosPro and Ultra Fractal and unfortunately only just recently realized I should be saving my parameters!

Anyhow, I've tried a few different things, including: varying the exponent with each iteration; cycling among two or three functions, each with a slightly different exponent; or trying a different function every nth iteration...

One thing that's been very helpful is finding an interesting spot on the Mandelbrot, then switching to the corresponding Julia, then changing the parameters to see if something interesting results.

Has anyone else run across these interesting geometric patterns?  I like them because they're a combination of angles, shapes, and curves.  They remind me of intricate carpet designs.

Linkback: https://fractalforums.org/index.php?topic=4592.0

Offline julofi

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Re: Very Slight Variations on the Mandelbrot Exponent
« Reply #1 on: January 12, 2022, 10:39:04 AM »
In purple, a mandelbrot julia with an inverse plane transformation and weird exponents, a bit similar pattern.
Julia Constant: -0.9,0i; Exponent -1.79; Bailout 3.7; Iterations 135; Colouring: Distance
The second one is more like what you say: Mandelbrot; exponent  2.01; Bailout 2; Iterations 1024; Colouring Escape. The transparent thingies are 2 orbit traps.
Both made with FFExplorer time ago.

Offline hobold

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Re: Very Slight Variations on the Mandelbrot Exponent
« Reply #2 on: January 12, 2022, 11:24:06 AM »
Some of these effects look like caleidoscopic patterns. So they are likely caused by the inevitable "cut" (a discontinuity of sorts) which ultimately stems from inherent limitations of the complex logarithm. (Integer exponents successfully hide these limitations, but all other exponents must expose them.)

Usually that cut is placed along the negative real axis. But for artistic purposes, it should be possible to move it around to be any ray from the origin in any arbitrary direction. (Actually there is even more artistic freedom here.)

So the main antenna is merely the default position; it should be possible to to move the newly introduced details to other places of the original fractal.

Offline youhn

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Re: Very Slight Variations on the Mandelbrot Exponent
« Reply #3 on: January 12, 2022, 11:19:35 PM »
To me it looks like the Burning Ship fractal. Some examples attached.

Offline BrokenParameter

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Re: Very Slight Variations on the Mandelbrot Exponent
« Reply #4 on: January 13, 2022, 02:05:43 AM »
Wow, thanks for your responses!  Those are great patterns.

Yes, there are definitely discontinuities going on.  If I zoom in, I can see that the patterns result from fractures, and sometimes the lines and shapes join up unevenly.

The Burning Ship examples are very similar to the things I've seen.  I've also noticed that the Burning Ship often looks skewed, as if you're looking down at the pattern from an angle.

Offline BrokenParameter

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Re: Very Slight Variations on the Mandelbrot Exponent
« Reply #5 on: Today at 04:37:43 AM »
Hello all!

Just wanted to share a new fractal I found today.

Formula is z = z^2.0000606 + c, except that every 125th iteration, it uses z = z^4.0001 + c.

Location is (-1.4366879028759731665,0) with magnification of 5.6161523E8.