Title: Mandelbrot Foam IIIa
Post by: Pauldelbrot on January 22, 2013, 11:31:55 PM
Mandelbrot Foam IIIa
(http://nocache-nocookies.digitalgott.com/gallery/13/511_22_01_13_11_31_55.jpeg)
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=13245
This colorful image has the same non-pixel-varying parameter as the Mandelbrot Foam II images, but a different initial z and initial w. It reveals the presence of budded mirrors, which in a one-variable complex mapping indicate the potential for Herman rings in the dynamics with suitable parameter choices. Here, I'm not sure what it means, but there are two of them visible here (and more, quite a few more, out of view to the sides).
(http://nocache-nocookies.digitalgott.com/gallery/13/511_22_01_13_11_31_55.jpeg)
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=13245
This colorful image has the same non-pixel-varying parameter as the Mandelbrot Foam II images, but a different initial z and initial w. It reveals the presence of budded mirrors, which in a one-variable complex mapping indicate the potential for Herman rings in the dynamics with suitable parameter choices. Here, I'm not sure what it means, but there are two of them visible here (and more, quite a few more, out of view to the sides).
Title: Re: Mandelbrot Foam IIIa
Post by: cKleinhuis on January 22, 2013, 11:50:21 PM
wow, that julia mirror looks really like ... dunno ... somethin ... spiritual!
the whole thing looks like a layered fractal, but there is some strange way of interaction ... nice and inspiring!
the whole thing looks like a layered fractal, but there is some strange way of interaction ... nice and inspiring!
Title: Re: Mandelbrot Foam IIIa
Post by: Pauldelbrot on January 23, 2013, 02:25:25 AM
Thanks!
No layers, though. That's how the fractal actually looks. :)
No layers, though. That's how the fractal actually looks. :)
Title: Re: Mandelbrot Foam IIIa
Post by: simon.snake on January 23, 2013, 12:03:23 PM
Hi Paul
Looks like a really lovely fractal.
Is it possible to generate this fractal in FractInt or Fractal eXtreme?
I know FractInt has an inbuilt formula parser, I just haven't seen your formula.
FX has to have plugins written to create additional formulae, but I haven't the skills or the necessary compiler available to do so.
Simon
Looks like a really lovely fractal.
Is it possible to generate this fractal in FractInt or Fractal eXtreme?
I know FractInt has an inbuilt formula parser, I just haven't seen your formula.
FX has to have plugins written to create additional formulae, but I haven't the skills or the necessary compiler available to do so.
Simon
Title: Re: Mandelbrot Foam IIIa
Post by: Pauldelbrot on January 23, 2013, 04:55:49 PM
It should be possible in Fractint, though you'll be much more limited in what you can do with coloring, and I don't know if deepzooming will work with it.
Title: Re: Mandelbrot Foam IIIa
Post by: rollercoaster158 on January 25, 2013, 03:44:03 AM
Excellent work! Have you gotten this one to work on Ultrafractal yet?
Title: Re: Mandelbrot Foam IIIa
Post by: Pauldelbrot on January 25, 2013, 04:54:32 AM
Already ahead of you there. The thread in the UF forum about "experimental quadratic two variable" somethingorother has a formula. :)
Title: Re: Mandelbrot Foam IIIa
Post by: fractalchemist on January 25, 2013, 11:59:01 AM
Paul what are "Herman rings", can you explain for my non math brain?
Title: Re: Mandelbrot Foam IIIa
Post by: AtomicZagnut on January 25, 2013, 12:18:13 PM
Amazing that it's not layered, as it really does look that way. Excellent work!
I, too, would also like to know more about Herman rings.
I, too, would also like to know more about Herman rings.
Title: Re: Mandelbrot Foam IIIa
Post by: fractalchemist on January 25, 2013, 12:25:32 PM
Formula is in the Ultra Fractal database in EM.ufm , as Pauldebrot.
Use smooth coloring.
You can also use Paul's Multiwave coring....also for UF.
http://www.fractalforums.com/ultrafractal/multiwave-coloring-for-mandelbrot/
Use smooth coloring.
You can also use Paul's Multiwave coring....also for UF.
http://www.fractalforums.com/ultrafractal/multiwave-coloring-for-mandelbrot/
Title: Re: Mandelbrot Foam IIIa
Post by: Alef on January 25, 2013, 03:21:16 PM
You get interesting fractal you get when z seed=0.0001, w seed is some very small nonzero real number like 0.00000000000000001 and multiplication factor is 0.5 - 1.
Then shape is like of Mandelbrot, but bulbs are like sunflowers with julia like sets as seeds, each with different ornaments.
Actualy shapes are less so interesting than this, but then it have interesting properties. W seed =0 represents singularity generating standart mandelbrot. So then its some system aproaching 1/infinity. You can alsou do burning ship from this.
Formula is here:
http://www.fractalforums.com/ultrafractal/complex-2-variable-quadratic-experiment/ (http://www.fractalforums.com/ultrafractal/complex-2-variable-quadratic-experiment/)
Then shape is like of Mandelbrot, but bulbs are like sunflowers with julia like sets as seeds, each with different ornaments.
Actualy shapes are less so interesting than this, but then it have interesting properties. W seed =0 represents singularity generating standart mandelbrot. So then its some system aproaching 1/infinity. You can alsou do burning ship from this.
Code:
Flores_Helianthi {
fractal:
title="Flores_Helianthi" width=640 height=480 layers=1
credits="Edgar;1/25/2013;Edgar;1/24/2013"
layer:
caption="Background" opacity=100
mapping:
center=-0.5/0 magn=1
formula:
maxiter=500 percheck=off filename="em.ufm" entry="Pauldebrot"
p_seed=0.0001/0 p_seed2=0.00000000001/0 p_wfactor=0.6
p_bailout=1000000.0 p_settype=Mandelbrot p_switchsettype=Julia
p_julia=0.3/0.6 f_func_post=ident p_zfunction=None p_inverted=no
inside:
transfer=none
outside:
transfer=linear filename="em.ucl" entry="Pauldebrots_Smooth"
p_convergent=no p_esm=no p_alt=no p_perfix=1.0 p_zmax=100 p_power=2
p_fit=yes p_fittimes=1.0 p_fitminit=1 p_fitmaxit=500 p_transfer=Log
p_transpower=3.0 p_bailout=1000000.0
gradient:
smooth=yes index=0 color=8716288 index=100 color=16121855 index=200
color=46591 index=300 color=156
opacity:
smooth=no index=0 opacity=255
}
Formula is here:
http://www.fractalforums.com/ultrafractal/complex-2-variable-quadratic-experiment/ (http://www.fractalforums.com/ultrafractal/complex-2-variable-quadratic-experiment/)
Title: Re: Mandelbrot Foam IIIa
Post by: Pauldelbrot on January 25, 2013, 06:57:45 PM
Amazing that it's not layered, as it really does look that way. Excellent work!
I, too, would also like to know more about Herman rings.
I, too, would also like to know more about Herman rings.
Like a Siegel disk, except hollow. The normal M-set can't produce them, nor any polynomial map of one complex variable, but some rational maps of one complex variable can, of which Supernova is one such map:
If b = 0, a is on the unit circle with irrational angle, and c is pure real or pure imaginary and |c| > 3, Herman rings tend to occur in the dynamic plane. The images aren't hugely fascinating compared to the parameter space images from the same map, though, particularly c-plane images with suitable choices of b and a. There you find "bud mirror" lines on the axes and sometimes other "bud mirror" arcs elsewhere; the dynamic plane has Herman rings for irrational points (as defined ... somehow. "Not exactly at any bud-pair's base", I think) on these mirrors. Elsewhere it has up to four attractors, three finite and one superattracting unmoving one at infinity. The finite ones depend on the parameters and if b = 0 one is stuck at zero and also superattracting, regardless of a and c. Images of all slices contain seahorse or other shapes that have been hollowed, with points going to infinity surrounding a lake of points going to zero or vice-versa; changing b away from 0 moves the latter attractor off zero, makes it less stable (or even unstable, bifurcating or simply ceasing to exist), and makes the "goes to zero" regions gnarl, distort, fragment, or convert into thick foam. Moving b into a b-plane minibrot really does interesting things to those regions.
There are images, mostly parameter plane, here; search the gallery for images tagged "supernova" (all the ones by me, and probably some of the others) or go to these big zoom sequences in parameter space (http://www.fractalforums.com/index.php?action=gallery;sa=search2;key=Supernova++seahorse) some of which show some of the effects of moving b off 0.