• May 18, 2021, 04:32:10 PM

### Author Topic:  Juliter Transformation  (Read 1285 times)

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#### C0ryMcG

• Posts: 263
##### Re: Juliter Transformation
« Reply #30 on: April 20, 2021, 07:00:09 AM »
You can also interleave the Juliper stuff or however you call it with the main fractal.
Here's an example of a Julia set of z^2 + c, where c starts out as whatever seed you select, then changes at each Julia set iteration according to the Z^2 + c, z_0 = 0 iteration. So if c_0 is inside, the c values in the Julia like set jumps on a period, and you get a normal Julia set only if the period is 1.

Here's an example.
This is a fun technique... It could apply to any fractal that follows the idea that z0=pixel position, and I like that it makes julia-like images that have a bit more variety in the structures than the usual ones do... I get a bit tired of how very self-similar Julias can be.

#### pauldelbrot

• 3f
• Posts: 2790
##### Re: Juliter Transformation
« Reply #31 on: April 20, 2021, 07:46:54 AM »
If it can be made compatible with distance estimator (and that's probably as simple as switching which derivative formula is being used at the appropriate iteration) then it can be applied also to ABox, ASurf, Mandelbulb, and other 3D fractals that have Julia and Mandelbrot modes.

#### gerrit

• 3f
• Posts: 2407
##### Re: Juliter Transformation
« Reply #32 on: April 22, 2021, 07:56:54 AM »
These extended Julia sets (by making the seed vary over its orbit) are more often disconnected than the normal ones. The interiors look different, here's one with a recursive dogfight shape not existing in normal Julia sets (of z^2 + c).

#### pauldelbrot

• 3f
• Posts: 2790
##### Re: Juliter Transformation
« Reply #33 on: April 22, 2021, 03:33:41 PM »
It's equivalent to iterating a more complex function. For instance, if you alternately use c and d as seeds, you're effectively iterating (z2 + c)2 + d, a two-parameter quartic polynomial.

#### gerrit

• 3f
• Posts: 2407
##### Re: Juliter Transformation
« Reply #34 on: April 22, 2021, 04:19:32 PM »
It's equivalent to iterating a more complex function. For instance, if you alternately use c and d as seeds, you're effectively iterating (z2 + c)2 + d, a two-parameter quartic polynomial.
Of course anything is a function but it's not a polynomial in general. We have Julia-like iterations with c changing each iteration according to the critical orbit of c_0, and that orbit is finite only at a set of points of measure zero, so only in those special circumstances (when the orbit is  (pre)periodic) is the extended Julia set the normal Julia set of some higher order polynomial.

#### pauldelbrot

• 3f
• Posts: 2790
##### Re: Juliter Transformation
« Reply #35 on: April 22, 2021, 05:10:03 PM »
In that case we're into the realm of 2-variable quadratics, the realm inhabited by things like enigma and mandelbrot foam.

$w \leftarrow w^2 + c$
$z \leftarrow z^2 + w$

with c a parameter, w initially 0, and z the pixel coordinate for a Julia like fractal. In this case there's no feedback from z to w, unlike mfoam and enigma.

#### gerrit

• 3f
• Posts: 2407
##### Re: Juliter Transformation
« Reply #36 on: April 22, 2021, 07:34:56 PM »
In that case we're into the realm of 2-variable quadratics, the realm inhabited by things like enigma and mandelbrot foam.

$w \leftarrow w^2 + c$
$z \leftarrow z^2 + w$

with c a parameter, w initially 0, and z the pixel coordinate for a Julia like fractal. In this case there's no feedback from z to w, unlike mfoam and enigma.
Right, here's my UF code; first select c using "switch feature".
Code: [Select]
Mbrot {init:   c = #pixelloop:   z = z^2 + cbailout:   |z| < @bailoutdefault:  title = "Mbrot"  center = (0, 0)  float param bailout    caption = "bailout"    default = 1e10  endparam  switch:    type = "XJulia"    seed = #pixel    bailout = bailout}XJulia {init:   c = @seed   z = #pixelloop:   w = w^2 + c   z = z^2 + wbailout:   |z| < @bailoutdefault:  title = "XJulia"  center = (0, 0)  float param bailout    caption = "bailout"    default = 1e10  endparam  complex param seed    caption = "seed"    default = 0     endparam}

#### gerson

• Fractal Frankfurter
• Posts: 609
##### Re: Juliter Transformation
« Reply #37 on: April 26, 2021, 08:48:58 PM »
Posted another mandelbrot with juliter option
https://fractalforums.org/index.php?topic=3800.msg28182#msg28182

#### Alef

• Fractal Friar
• Posts: 140
• a catalisator / z=z*sinh(z)-c^2
##### Re: Juliter Transformation
« Reply #38 on: April 27, 2021, 06:17:06 PM »
If it can be made compatible with distance estimator (and that's probably as simple as switching which derivative formula is being used at the appropriate iteration) then it can be applied also to ABox, ASurf, Mandelbulb, and other 3D fractals that have Julia and Mandelbrot modes.
With Mandelbulb3D I made number of simmilar fractal hybrids but useing different formulas and julia c = (0,0).
First iterations made big shape and then folowing iterations created small shapes. (Big fractal consisting of many same shapes of smaller fractal). Like iterating Abox or Asurf fractal for some 7 iterations and then menger sponge fractal (wich coresponds to julia set c = 0,0). - Asurf mushroom consisting of small mengers.
It was not the same as this thing but that sounds perfectly possible.
Mandelbulb3D have it's own version of DE, it fully works without derivatives. And I don't know how it calculates DE in fractal hybrids.
a catalisator / z=z*sinh(z)-c2

#### julofi

• Fractal Furball
• Posts: 247
##### Re: Juliter Transformation
« Reply #39 on: April 30, 2021, 04:11:20 PM »
I am always more interested in the artistical possibilies of software than in the maths (that I don't understand).
Anyway, after some weeks without doing fractals, I made one just today. I have used the last version of FFExplorer and its Juliter transformation, but I have applied old features too, like using any image as orbital trap in fractals.
The fish in the fractal is a png image downloaded from a free clipart.
The fractal itself is obviously a lambda. Raw image, no post-processing.

#### gerson

• Fractal Frankfurter
• Posts: 609
##### Re: Juliter Transformation
« Reply #40 on: May 03, 2021, 01:34:09 AM »
@julofi beautiful image. It remembered me raimuraito program by Yasuo Kamei. It use fish as an orbit trap option.

#### julofi

• Fractal Furball
• Posts: 247
##### Re: Juliter Transformation
« Reply #41 on: May 03, 2021, 12:17:30 PM »
I know about Raimuraito since years ago. It is a software that I like but that I barely use because its resolution is limited to 1024*768 and that is too low for me. I have used a lot the kilt tool it includes to make fractal-tiles.
I like the way Raimuraito allows you to configure many aspects of the orbit trap, for example, if you choose the trap "flower" you have a lot of options to change that flower: number of petals, length, size of center, etc. I have told SergioCT many times that he should include something like that in FFExplorer, to change for example, the number and lenght of the star type orbit traps, the shape of points of the stars, etc.

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