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Author Topic:  Nested Herman rings  (Read 297 times)

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Offline xenodreambuie

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Nested Herman rings
« on: December 03, 2020, 11:41:27 PM »
This paper has a quartic formula with nested Herman rings (shown in Figure 2).
"On the formulas of meromorphic functions with periodic Herman rings"
https://arxiv.org/abs/2011.10935

I have not succeeded in reproducing the structure in Figure 2.
It's a nice formula with some interesting Julia sets anyway, but I'd like to know what's necessary to get that result. No matter what settings I use, half of the holes tend to escape and half tend to converge, which doesn't tally with their result.

Edit: from looking at the proof on page 12, it appears that there is a missing minus sign in the exponential in the denominator in the formula above Figure 2.

Linkback: https://fractalforums.org/programming/11/nested-herman-rings/3898/
« Last Edit: December 04, 2020, 12:06:16 AM by xenodreambuie, Reason: Found the problem? »

Offline FractalAlex

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Re: Nested Herman rings
« Reply #1 on: December 04, 2020, 12:22:31 AM »
Interesting... These Herman rings are very intriguing. Especially figure 5... I wonder if the abs() variants (Burning Ship, Buffalo, Celtic, Simonbrot, etc.) exist...
"I am lightning, the rain transformed."
- Raiden, Metal Gear Solid 4: Guns of the Patriots

Offline xenodreambuie

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Re: Nested Herman rings
« Reply #2 on: December 04, 2020, 01:30:23 AM »
Interesting... These Herman rings are very intriguing. Especially figure 5... I wonder if the abs() variants (Burning Ship, Buffalo, Celtic, Simonbrot, etc.) exist...

What do you mean by exist? You can add abs variants to practically any formula, usually to z before the rest of the function, but also useful in the middle of some formulas. I've already added it to my implementation of this quartic function, and it gives some good results. But note that it destroys actual Herman ring structures.

Offline xenodreambuie

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Re: Nested Herman rings
« Reply #3 on: December 04, 2020, 04:24:03 AM »
With the fixed formula, a slight change in angle and careful adjustment of low bailout I can get this image, which satisfies my curiosity. This is the Julia set with c=0. Blue is convergent and white is boundary. There is an asymmetry between alternate rings that I suspect is due to criticality and associated precision issues.

Offline FractalAlex

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Re: Nested Herman rings
« Reply #4 on: December 04, 2020, 09:38:12 AM »
Excellent work. I'd love to have a formula file for this one.  :thumb_up:

Offline xenodreambuie

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Re: Nested Herman rings
« Reply #5 on: December 04, 2020, 10:49:16 AM »
Damn, I had a typo in my code, and the formula above Figure 2 in the paper is correct, though it doesn't quite tally with the one used in the proof. Now I get the correct picture, and it does in fact have alternate holes escaping and converging.

Offline xenodreambuie

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Re: Nested Herman rings
« Reply #6 on: December 05, 2020, 08:46:42 AM »
@FractalAlex: about Abs() variants. As I said, you can add abs to any formula, but for general functions that aren't pure powers, the Mandelbrot forms are not pretty and tend to look similar. The main use for them is navigating to interesting Julia sets. These are generally found in the holes in the set or near the edges. Here's a Mandelbrot set of the quartic function with the standard abs(z), for example, with the angle parameter at 123 degrees.

Offline FractalAlex

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Re: Nested Herman rings
« Reply #7 on: December 05, 2020, 11:10:20 AM »
Nice!


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