Unveiling the Fractal Structure of Julia Sets with Lagrangian Descriptors

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Offline Adam Majewski

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« on: March 26, 2020, 10:13:30 AM »
https://arxiv.org/abs/2001.08937

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/unveiling-the-fractal-structure-of-julia-sets-with-lagrangian-descriptors/3376/

Offline pauldelbrot

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« Reply #1 on: March 27, 2020, 04:15:24 AM »
Code: [Select]
DLD {
; Based on https://arxiv.org/pdf/2001.08937.pdf
init:
  float sum = 0.0
  float lastx = 0.0
  float lasty = 0.0
  float lastz = 0.0
  int i = 0
loop:
  float d = |#z|
  float dd = 1/(d + 1)
  float xx = 2*real(#z)*dd
  float yy = 2*imag(#z)*dd
  float zz = (d - 1)*dd     ; Riemann sphere coordinates
  IF (i > 0)
    sum = sum + (xx - lastx)^@power + (yy - lasty)^@power + (zz - lastz)^@power
  ENDIF
  i = i + 1
  lastx = xx
  lasty = yy
  lastz = zz
final:
  #index = sum/(i - 1)
default:
  title = "Discrete Langrangian Descriptors"
  param power
    caption = "Power"
    default = 0.25
    hint = "Affects the steepness of the gradient near singularities (fractal features like a Julia set)"
    min = 0.0
  endparam
}

Offline gerrit

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« Reply #2 on: March 27, 2020, 05:08:25 AM »
https://arxiv.org/abs/2001.08937
A coloring method disguised as a math paper. Looking forward to seeing what it looks like.

Offline gerrit

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« Reply #3 on: March 27, 2020, 07:07:29 AM »
Code: [Select]
[/quote]
Thanks for the code. Here's a familiar Julia set with Lagrangian method power=0.001.

Offline Adam Majewski

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« Reply #4 on: March 27, 2020, 06:40:33 PM »
Code: [Select]
[quote author=gerrit link=topic=3376.msg19814#msg19814 date=1585289249]
Thanks for the code. Here's a familiar Julia set with Lagrangian method power=0.001.
[/quote]

can you post other examples from the paper and your code ?


Offline gerrit

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« Reply #6 on: March 27, 2020, 10:42:35 PM »
can you post other examples from the paper and your code ?
Pauldelbrot posted code, and I am not so interested in the paper examples.
Here's a scene from Mandelbrot foam with this coloring.

Offline Adam Majewski

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« Reply #7 on: March 28, 2020, 09:28:01 AM »
Pauldelbrot posted code ...

so you use code by Pauldebrot and make images in Ultrafractal ?







Offline sjhalayka

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« Reply #8 on: March 28, 2020, 05:49:03 PM »
Pardon me Adam, but before I scour your paper, I am wondering why you use the name Lagrangian? Do you use kinetic and potential energy?

Offline gerrit

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« Reply #9 on: March 28, 2020, 07:45:52 PM »
Julia set of \( z^2+i \) rendered with power 20, 2, 0.2, 0.02 in normal reading order.
5th image replaces \( x^{power} \) with \( log(abs(x)) \) in the formula.

Offline pauldelbrot

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« Reply #10 on: April 01, 2020, 03:37:31 PM »
DLD coloring applied to a) a Triskelion Julia set with two Siegel disks (light and dark blue, with conjugate rotations related to the golden mean) and a parabolic basin of period 2; b) a slice of the Triskelion parameter space. The latter shows that this coloring will bring out details in the "spacefilling" zones where no attractors exist, and I can confirm it will show structures in spacefilling Julia sets from parameters in such zones. It also, of course, doodles interesting designs into the bulbs, minibrots, and other structures where one or more attractors are stable.

Offline Adam Majewski

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« Reply #11 on: April 01, 2020, 04:29:09 PM »
Pardon me Adam, but before I scour your paper, I am wondering why you use the name Lagrangian? Do you use kinetic and potential energy?
I think that best explanations is in the paper. ( I'm not lazy, simply I do not know (:-))

Offline pauldelbrot

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« Reply #12 on: April 02, 2020, 07:59:14 AM »
Supernova parameter space, c-plane, b = 0 and a = eiΦ, where Φ is the golden mean minus one.

Offline gerrit

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« Reply #13 on: April 02, 2020, 09:09:47 AM »
Another Mandelbrot foam using it also for the interior.

Offline pauldelbrot

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« Reply #14 on: April 05, 2020, 04:00:58 AM »
Matchmaker parameter space.


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