• November 12, 2022, 08:11:04 AM

Author Topic:  Juliter Transformation  (Read 2396 times)

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Offline sergioct

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Juliter Transformation
« on: April 05, 2021, 02:29:53 PM »
I have been working on a transformation consisting of altering the behaviour of the Mandelbrot algorithm, in such a way that "C" acquires the value of a constant "J" in a given iteration "I". At that moment, the algorithm will stop calculating the Mandelbrot Set and will use the rest of the iterations to calculate the Julia Set, which will be embedded within the Mandelbrot calculated so far and influenced by the value of "Z" after those first iterations.

My tests seems to indicate that the transformation performs well for any function handled by the Mandelbrot method, so it could be a good starting point to complicate it and do cooler things ;)

The most suitable name that I have come up with for them is "Juliter Transformation", and I have tried to explain it to my students in this document: https://fractalfun.es/juliter.php?i=en

I leave you here some of my results.



Linkback: https://fractalforums.org/index.php?topic=4148.0

Offline gerrit

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Re: Juliter Transformation
« Reply #1 on: April 05, 2021, 03:25:54 PM »
I think another way of saying it is you warp a picture by sending each pixel p to f(p) with f(p) = I iterations of z^2 + p, z0 =0 (or any other fractal), and your picture happens to be a Julia set.

Offline gerrit

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Re: Juliter Transformation
« Reply #2 on: April 05, 2021, 06:27:48 PM »
Had some fun with this transform. I applied it with I = 2500 to the Mandelbrot set itself, see below for cusp of period 3 mini normal and then after the Juliter transform, and then the largest bulb left of the bulb on top. UF code for transform:
Code: [Select]
JuliterTransform {
; from @sergioct
transform:
  complex func f(const complex w, int it)
     z = 0
     int k = 0
     while (k<it)
       z = z^2 + w
       if(|z|>4)
         z = 1e12
         return z
       endif
       k = k + 1
     endwhile
     return z
  endfunc
  if @isOn
    #pixel = f(#pixel,@it)
  endif
default:
  title = "JuliterTransform"
  bool param ison
    caption = "On"
    default = false
    hint = "Turn on/off"
  endparam
  int param it
    caption = "I"
    default = 16
  endparam
}

Offline sergioct

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Re: Juliter Transformation
« Reply #3 on: April 05, 2021, 08:48:19 PM »
Hi @gerrit,

Do you think your rethinking could produce an image equivalent to the ones I show in the examples?

Quote
I think another way of saying it is you warp a picture by sending each pixel p to f(p) with f(p) = I iterations of z^2 + p, z0 =0 (or any other fractal), and your picture happens to be a Julia set.

I would like to know if both agorithms are equivalent, but I don't understand the UF syntax well. It would be very good if it were, since your adaptation seems like a generalization to "mix" any fractal with any other. Although I have my doubts about whether the properties that I describe in the document would be maintained, since I believe that they are due to the close relationship between the Mandelbrots and their Julias.

These are the parameters of the following image:

Iterations = 256
J = (-0.447, -0.564)
I = 16

Offline sergioct

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Re: Juliter Transformation
« Reply #4 on: April 05, 2021, 08:53:21 PM »
On the right you can see the Mandelbrot Set with the Juliter Transformation that corresponds to the Julia Set on the left.


Offline gerrit

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Re: Juliter Transformation
« Reply #5 on: April 05, 2021, 11:04:50 PM »
Hi @gerrit,

Do you think your rethinking could produce an image equivalent to the ones I show in the examples?

I would like to know if both agorithms are equivalent, but I don't understand the UF syntax well. It would be very good if it were, since your adaptation seems like a generalization to "mix" any fractal with any other. Although I have my doubts about whether the properties that I describe in the document would be maintained, since I believe that they are due to the close relationship between the Mandelbrots and their Julias.

These are the parameters of the following image:

Iterations = 256
J = (-0.447, -0.564)
I = 16
Looks the same (first image). Increasing I from 16 to a large value gives cleaner result (2nd image).

Offline sergioct

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Re: Juliter Transformation
« Reply #6 on: April 06, 2021, 08:17:21 AM »
Quote
Looks the same (first image). Increasing I from 16 to a large value gives cleaner result (2nd image).

Yes, your second result also matches mine by increasing "I" :)

Offline gerrit

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Re: Juliter Transformation
« Reply #7 on: April 06, 2021, 09:13:58 PM »
I thought I had seen this before.

See https://fractalforums.org/index.php?topic=442.msg2204#msg2204 where this method amongst others is discussed more generally.

Online xenodreambuie

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Re: Juliter Transformation
« Reply #8 on: April 07, 2021, 04:33:47 AM »
I thought I had seen this before.

See https://fractalforums.org/index.php?topic=442.msg2204#msg2204 where this method amongst others is discussed more generally.

Chris M Thomasson (in FB and before that, Google+) has also been doing the Mandelbrot transformation before Julia for some time, and calls it Mulia.

The general principle of pre-transforms seems the same as what is used for water effects, gnarls and other filters. I hadn't thought of doing the Mandelbrot hybrid this way rather than with some integrated method; thanks!

Offline sergioct

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Re: Juliter Transformation
« Reply #9 on: April 07, 2021, 09:18:14 AM »
For a few days I have been contrasting results with Chris M Thomasson on FB and they are very similar.

Offline superheal

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Re: Juliter Transformation
« Reply #10 on: April 08, 2021, 09:41:43 AM »
I tried changing the function to newtons iterations (f(z) = z^3 - 1) for iterations >= 8
I also had to change the bailout criterion from escaping to converging (|z - zprev| < 1e-6)

Offline Alef

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Re: Juliter Transformation
« Reply #11 on: April 09, 2021, 06:04:56 PM »
Formula sugestion N1
Maybe try z=z*sinh(z)-c2 (or more generalised form  z=z(n-1)*sinh(z)-cn but higher powers are boring)
It alredy have a julia set like satelites near spiral zooms at the every level of magnification and alsou inside satelites.

Interesting why -C2 brings back good old mandelbrot (lambda) shape with z*sinh(z) when + C don't. There must be some math behind this equation I don't understand.
« Last Edit: April 15, 2021, 07:14:41 PM by Alef »
by Edgar Malinovsky aka Edgars Malinovskis.

Offline gerson

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Re: Juliter Transformation
« Reply #12 on: April 12, 2021, 06:19:10 PM »
A mandelbrot using juliter on FFExplorer. With juliter on left.
« Last Edit: April 14, 2021, 07:25:17 PM by gerson »

Offline gerson

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Re: Juliter Transformation
« Reply #13 on: April 12, 2021, 06:19:59 PM »
A detail with other juliter iteration.

Offline gerson

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Re: Juliter Transformation
« Reply #14 on: April 12, 2021, 06:23:18 PM »
This is a phoenix mandelbrot inverse. The juliter are the most part of the image.


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