The Mandelbrot Set inside-out or unfurled and laid flat...

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Offline JonathanD

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« on: February 03, 2020, 07:16:58 PM »
Howdy folks,

I have another interesting twist to offer you, with an algorithm that lets me unroll the Mandelbrot Set around any point, by plotting successively larger circles of points in M as rows of pixels.  This is what you see when every circle of pixels in a progression of rings around that location is taken apart at a point and laid flat.  So what is at the top is highly-magnified near the cusp while what is on the bottom row is zoomed-out to the edge of the entire Set.  The aspect ratio was chosen because the width is about 2Pi times the height.  One can choose any location, magnification, or angle of origin and rotation, using this method.  But it is not something I have seen elsewhere.  I'll post more examples later.

Enjoy,

Jonathan


Linkback: https://fractalforums.org/share-a-fractal/22/the-mandelbrot-set-inside-out-or-unfurled-and-laid-flat/3303/
Jonathan J. Dickau


Offline gerrit

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« Reply #2 on: February 03, 2020, 08:40:54 PM »
And this is what that M(3,1) Misiurewicz point looks like at 1E10 magnification (bottom) use the exponential map.


Offline Adam Majewski

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« Reply #4 on: February 04, 2020, 04:09:29 PM »
And this is what that M(3,1) Misiurewicz point looks like at 1E10 magnification (bottom) use the exponential map.

Cool
Can you describe more the algorithm?
Can you post your image to commons ? ( sorry if I always ask for it )

Offline gerrit

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« Reply #5 on: February 04, 2020, 08:26:31 PM »
Cool
Can you describe more the algorithm?
Can you post your image to commons ? ( sorry if I always ask for it )
Algorithm has been discussed here before, but below my the Ultra Fractal transform code.
Feel free to use image as you please, I'm not into the commons thing.

Code: [Select]
ExponentialMap {
;
; Vertical exponential transform.
; Make narrow high window (say 100X800), zoom, put point of interest at bottom.
; Turn on. Tweak with the 2 controls, and select appropriate width (depends on image,
; narrower for deeper zooms).
;
global:
  if (4 * #height < 3 * #width)
    pixeldim = 3/#magn/#height
  else
    pixeldim = 4/#magn/#width
  endif
  w = #width * pixeldim
  h = #height * pixeldim
  cc = #center
  c0 = cc - 1i/2 * h +0i*w/2 -w/4 * @sh
  b = w/h*log(#magn) *@b
  a = 1i*w/b
transform:
    if @ison
      c = #pixel
      dc = a*(exp(-1i*b/w*(c-c0)))
      #pixel = c0+dc
    endif
default:
  title = "Exponential map"
  param ison
    caption = "On"
    default = false
  endparam
  float param b
    caption = "vert. control"
    default = 1.2
  endparam
  float param sh
    caption = "hor. shift"
    default = 0
  endparam
}


Offline Adam Majewski

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« Reply #6 on: February 04, 2020, 10:05:23 PM »
Algorithm has been discussed here before, but below my the Ultra Fractal transform code.
Feel free to use image as you please, I'm not into the commons thing.
Do I have your permission to publish it under :
https://creativecommons.org/licenses/by-sa/4.0/deed.en
licence?

Offline gerrit

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« Reply #7 on: February 04, 2020, 10:58:58 PM »

Offline JonathanD

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« Reply #8 on: February 05, 2020, 06:19:34 AM »
Wow that's way cool Gerrit, and thanks greatly to Adam...

But what I have rendered is not the same in terms of the scaling; I think.  The steps drawing circles in my program are evenly spaced, then these are turned into rows of pixels which I believe produces a rotational to orthogonal translation with geometric or parabolic scaling not exponential.  I'll have to work this out or look at some spots both ways and compare.

I have some images where this technique is applied at branching Misiurewicz points.  Fun stuff!  Thanks for all the input and feedback.

More later, JJD

Offline gerrit

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« Reply #9 on: February 05, 2020, 06:24:35 AM »
Wow that's way cool Gerrit, and thanks greatly to Adam...

But what I have rendered is not the same in terms of the scaling; I think.  The steps drawing circles in my program are evenly spaced, then these are turned into rows of pixels which I believe produces a rotational to orthogonal translation with geometric or parabolic scaling not exponential.  I'll have to work this out or look at some spots both ways and compare.
Whatever you do can be stated as \( z \leftarrow z^2 +f(c) \) with c mapped to pixels usual way.
So what is your f(c)? Or maybe it has no explicit form?

Offline Adam Majewski

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« Reply #10 on: February 05, 2020, 03:41:18 PM »
Sure.

https://commons.wikimedia.org/wiki/File:M(3,1)_Misiurewicz_point_looks_like_at_1E10_magnification_(bottom)_use_the_exponential_map.jpg

done

Can you describe algorithm in simple words? ( here is along vertical line )
Can Exponential map be aplied along any direction , any curve ?

Offline JonathanD

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« Reply #11 on: February 05, 2020, 04:15:15 PM »
This is way cool...

But I don't know there is an easy way to obtain the kind of pixel mapping I am doing by fiddling with the formula and adding an f(x) term.  What I use is a brute force method or 1 to 1 relation, by first finding a circle of points around a given location, then using the standard Mandelbrot formula to calculate with no adjusted or varying terms.  Then it plots those circles as rows of pixels.

I think that means it is a linear relation since the formula for the periphery of a circle is 2Pi*r, and r varies in even increments.  Munafo and others use the standard pixel mapping.  But in my program; it is the OUTER LOOP that is reconfigured, while the inner loop (that does the calculations) remains unchanged.  It was written in Turbo Pascal ages ago, but if I can find the actual code I'll post that and others can translate.

I'll post a few more sample images here.

Regards, JJD

Offline gerrit

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« Reply #12 on: February 05, 2020, 06:05:51 PM »
https://commons.wikimedia.org/wiki/File:M(3,1)_Misiurewicz_point_looks_like_at_1E10_magnification_(bottom)_use_the_exponential_map.jpg

done

Can you describe algorithm in simple words? ( here is along vertical line )
Can Exponential map be aplied along any direction , any curve ?
As I said this has been discussed before, a keyword search shows I posted this code before:
https://fractalforums.org/ultrafractal/59/exponential-map-transform/1697/msg8580#msg8580
and all links back from that seem to lead to Claude, not sure if he has a blog post.
I forgot exactly how and why it works.

Offline Adam Majewski

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« Reply #13 on: February 05, 2020, 07:56:18 PM »
I forgot exactly how and why it works.

(:-)))

Offline gerrit

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« Reply #14 on: February 25, 2020, 06:10:34 AM »
Another cute one, journey past several minis.


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