• October 20, 2021, 03:55:43 AM

Author Topic: (Question) The minibrot in the heart?  (Read 262 times)

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Offline The_Blind_One

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(Question) The minibrot in the heart?
« on: October 04, 2021, 03:31:52 AM »
Ok so I remember seeing a video on youtube once (like 5 years ago lol) that showed that there was a minibrot hidden near the center of the major heart cardioid but I can't seem to find it again or whether I was just imagining things. It's either in the primary cardioid or in the heart of a different minibrot itself but I have no idea where that would be.

Am I imagining this or is there really a hidden minibrot in the black heart shape somewhere? :lol:

Any help would be greatly appreciated in answering this most dire of questions. 8)



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Offline C0ryMcG

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Re: The minibrot in the heart?
« Reply #1 on: October 04, 2021, 07:04:19 AM »
Are you describing a minibrot somehow existing as an island inside the main cartioid, not on the edge of it? If so, no, I believe it's been proven that with the standard mandelbrot formula perimeter is continuous, and that the cartioid contains only non-escaping points. So no possible islands.

Maybe something similar could be done with a different formula, though?

Offline The_Blind_One

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Re: The minibrot in the heart?
« Reply #2 on: October 04, 2021, 12:37:20 PM »
Yes that's what I meant. I must have been imagining things then. ^-^

Offline marcm200

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Re: The minibrot in the heart?
« Reply #3 on: October 04, 2021, 03:13:54 PM »
You can construct such a set, e.g. as a polynomial with a pre-orbit transcendental transformation of the pixel seed c

\[
f(z) := z^2+~\left(~{{14\cdot \arctan \left(10000000000\cdot \left| c\right| -1000000000
 \right)+8\cdot\pi\cdot c-7\cdot\pi}\over{8\cdot\pi}}~\right)
 \]

which basicaly copies the region around the period-3 minibrot at -1.75 into a circle around the origin with slight perturbations.


Offline The_Blind_One

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Re: The minibrot in the heart?
« Reply #4 on: October 04, 2021, 03:50:39 PM »
Thanks Marchman200 that's awesome! That math looks like magic to me though :yes:

I'm asking because I think I remember seeing someone doing a mandelbrot zoom and possibly finding a mandelbrot inside a mandelbrot's heart.

Looking at the buddhabrot from this video at the time segment indicated, it would also seem there's a lot more going on in the black area then one would think from a cursory glance at the shape.

https://youtu.be/9gk_8mQuerg?t=63

As you can see there's clearly another ''type'' of brot with a pointy head in there right at the center of that image.

Even more clearly depicted is on this website: https://benedikt-bitterli.me/buddhabrot/

There's a large animation there where he makes a buddhabrot transform back into the mandelbrot set and there's clearly more mandelbrots just plain hiding in the center of it.

https://benedikt-bitterli.me/buddhabrot/images/transform.mp4

I'm just your average idiot though :P
« Last Edit: October 04, 2021, 04:13:07 PM by The_Blind_One »

Offline youhn

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Re: The minibrot in the heart?
« Reply #5 on: October 04, 2021, 05:39:54 PM »
Per definition there are no minibrots in the pure mandelbrot rendered as escape time fractal. Period.

A Buddhabrot is a bit different, same basic formula but other way of calculating and coloring the grid. So naturally one would expect different results. Though there is no single buddhabrot. Some backgrounds taken from erleuchtet . org/2010/07/ridiculously-large-buddhabrot . html (replace " . " with "." to remove spaces for a functional link):

Quote
"The"  Buddhabrotbuddhabrot_symmetry.jpg

Since we still choose our samples randomly, every rendering will look different - depending on which orbits your random search finds. There is no "the" Buddhabrot (see Melinda's comment below - rendering for an infinite amount of time yields "the" Buddhabrot). This also means that the image is not symmetic on the small scale which is a good thing if you ask me. You could of course get a symmetric image of the same quality in half the time by mirroring it, but i think it's worth waiting for the asymmetric one. On the right is a part of the final rendering that exhibits superficial symmetry, but is composed of asymmetric orbits. Click it to view this detail in the final resolution.

If i had had my renderer run for much longer (maybe a week or a month), more and more orbits would have shown up, occluding each other and finally blurring the whole thing into the standard nebula-like buddhabrot. The key is to render just long enough until a sufficient amount of orbits have appeared and then stop.

Summarized into half a sentence from the link given by yourself:

"... have to sample many random trajectories and hope that they pass through the pixels we care about"

so with a bit of luck, the buddha can produce many shapes.

Btw, you can use software like Photoshop of The Gimp to simply paste a minibrot into the main cardioid. No math skills required.

Offline The_Blind_One

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Re: The minibrot in the heart?
« Reply #6 on: October 04, 2021, 08:42:57 PM »
If it's not in the main cardioid, it is perhaps possible that it exists inside another minibrot itself?

If nobody has heard or knows anything about it then it probably just something I imagined years ago. I just thought asking would give me a quicker answer since google didn't show me any results either even after a few hours search. Thanks everyone for helping answer me though. Much appreciated! :worship:

Offline claude

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Re: The minibrot in the heart?
« Reply #7 on: October 04, 2021, 09:08:23 PM »
The minibrots are solid too: there are conformal maps (a different one for each minibrot) between the cardioid-like shape and the unit disk, which covers the whole shape without leaving any gaps for extra minibrots.

You could construct a colouring algorithm that mapped say the radius 2 disk with the Mandelbrot set in it onto each minibrot cardioid (and disks too), and so on, if you wanted to go Full Recursive, but that'd require a fair bit more work than the regular M-set iterations.

See e.g. https://mathr.co.uk/misc/2013-04-02_moondelbrot.jpg for the Moon mapped to each component.

Offline marcm200

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Re: The minibrot in the heart?
« Reply #8 on: October 04, 2021, 09:19:22 PM »
You could find inner structure in multicritical polynomial parameter spaces when focusing e.g. on the number of critical orbits being bounded. Maybe you've seen something in that regard (heat-map colored).

Code: [Select]
(the first few terms in the series definition of cos(z)+c)
f(z)= c++(-0.0013888888888888889419+i*0)*z^6+(0+i*0)*z^5+(0.041666666666666664354+i*0)*z^4+(0+i*0)*z^3+(-0.5+i*0)*z^2+(0+i*0)*z+(1+i*0)

Offline lkmitch

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Re: The minibrot in the heart?
« Reply #9 on: October 06, 2021, 05:49:06 PM »
I suspect that this is not what the OP is thinking of, but here's an example of island minibrots inside of a main cardioid(-type) structure. The formula is:

z = ((z^2+c)/(z+c))^2.

If you speak Ultra Fractal, it's the Compound Mandelbrot formula in lkm3.ufm.

Offline The_Blind_One

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Re: The minibrot in the heart?
« Reply #10 on: October 09, 2021, 12:18:44 AM »
Thanks lkmitch for your suggestion.

That's pretty close what I had in mind actually.

I found a video which sortof describes what I meant as well but in this case it zooms in on the -1 circle instead of the main heart cardioid:

https://www.youtube.com/watch?v=AN5UH-xKG50&list=WL&index=197&ab_channel=Ben%27sFractals

(video is titled 'A normal mandelbrot set zoom' when it's clearly -not- a ''normal'' mandelbrot zoom lol)

In this video you can clearly see the guy zooming into the circle of the mandelbrot and generating a new area inside of it when properly zoomed in. I remember a video that did something similar to this but to the heart cardioid instead. I think it may have well been a similar video as this one that was using a different formula than the standard mandelbrot set to achieve this result and I just attributed it incorrectly in my memory. I'm just your average highschool idiot not a math magician lmao.

Thanks guys! I think we can consider this a case closed for now. There's no hidden heart mandelbrot hidden inside the standard mandelbrot set unless it is edited with magicians mathematical jutsu :))


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