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Author Topic: (Question) Why do mandelbrot sets resemble cycloids?  (Read 210 times)

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Offline DeusDarker

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(Question) Why do mandelbrot sets resemble cycloids?
« on: August 03, 2020, 02:23:34 AM »
Im really curious as to why they are structured as cycloids. the positive powered M-sets outlines structured after epicycloids and the negative powered M-sets are hypocycloids. does anyone have a decent explanation of why they are cycloidic in nature as well as why it goes epocyclic in the positive direction and hypocyclic in the negative direction?

Also, when zooming into the negative Mandelbrot Ive noticed something really odd,
1. they are very spongy in structure
2. as far as i have been able to tell there are no self similar smaller mini-sets inside the structure. it might very well be a deeper zoom is required, but with the positive M-sets its very easy to find a self similar scaled down version in the structure itself, not the case with the negative sets. all of them are just really spongy and as far as i zoom there are no self-similar structures (except for the sponge holes) but they do have spirals which is neat.

Thanks for your time, i hope someone can shed light on these questions.

Linkback: https://fractalforums.org/index.php?topic=3692.0

Offline C0ryMcG

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Re: Why do mandelbrot sets resemble cycloids?
« Reply #1 on: August 03, 2020, 06:49:54 AM »
I don't have many answers for you, but I do have more data!
One thing about formulas that invert, like anything where you'd use 1/z, escape coloring isn't going to get along well with it, because small values will quickly swap with big values on each iteration. One way I've tried in the past to get around that is to use every second iteration in my coloring calculations.
Another option is to simply use other types of coloring methods. If you've read many of my posts recently you'll know I'm a fan of Lagrangian Descriptors, described elsewhere on this forum, which reveals a lot of structure in formulas that are otherwise difficult to see any info in.

If you want more literal data, without requiring the image to look any good, I have an arrangement of purely analytical coloring methods that can give me good hints, and one I tried today was Destination Location. It calculates the attractor for a given pixel (or the closest in a cycle if it cycles) and returns a color based on the x/y coords.

Applying these to z^-2 + c shows me that this formula doesn't have minis that resemble the whole, just as you suspected, but it seems to be dominated by minis that resemble z^4+c. Strange!

I'll include renders of both Lagrangian coloring and Destination.

Offline DeusDarker

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Re: Why do mandelbrot sets resemble cycloids?
« Reply #2 on: August 03, 2020, 01:50:31 PM »
I don't have many answers for you, but I do have more data!
One thing about formulas that invert, like anything where you'd use 1/z, escape coloring isn't going to get along well with it, because small values will quickly swap with big values on each iteration. One way I've tried in the past to get around that is to use every second iteration in my coloring calculations.
Another option is to simply use other types of coloring methods. If you've read many of my posts recently you'll know I'm a fan of Lagrangian Descriptors, described elsewhere on this forum, which reveals a lot of structure in formulas that are otherwise difficult to see any info in.

If you want more literal data, without requiring the image to look any good, I have an arrangement of purely analytical coloring methods that can give me good hints, and one I tried today was Destination Location. It calculates the attractor for a given pixel (or the closest in a cycle if it cycles) and returns a color based on the x/y coords.

Applying these to z^-2 + c shows me that this formula doesn't have minis that resemble the whole, just as you suspected, but it seems to be dominated by minis that resemble z^4+c. Strange!

I'll include renders of both Lagrangian coloring and Destination.

Thank you so much for the feedback, its really appreciated! i had no idea the software i am using is so imprecise (XaoS 4.0) when it comes to detail. The coloring mode you mention looks really interesting, ill need to do some research on it and look at some of your older post. Unfortunately i am not too well versed in programming and i assume most fractal generators like yours are hand made home brew software so i might have to learn some programming if i want to get more presice detail, or maybe this fourm has some decent renders in the downloads section that can do Lagrangian Descriptors, again ill need to check. Thanks again!

Offline C0ryMcG

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Re: Why do mandelbrot sets resemble cycloids?
« Reply #3 on: August 03, 2020, 10:12:41 PM »
I am certainly using my own program. But I know there are downloadable programs that give color method options, although since Lagrangian Descriptors as a fractal coloring method originated from this forum not too long ago, I wouldn't expect many of them to have adopted that option, except the ones maintained by users from this forum.

The important thing, though, is that different methods will give you different visual data about the formulas. So even if you find a method unlike the ones I used, you'll still find something new to see.


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