New type of fractal

[jehovajah]

Maybe you could just show how it directly produces a Mandelbrot form to orient some of your readers. But the conformal mapping aspect is pretty fundamental if one wants to produce the holy grail from spherical balls! The Mandelbrot dynamic is not only beautiful, it is our entry point into describing many natural dynamic systems.

By the way, love your work on time scaled fractals. I think it is one of the most important fractal functions I've seen so far, especially in describing motion and growth in our universe.

Hey, but there are so many cool things fractals explain, it's no wonder great new leads get tossed aside for the shiny stuff! 

[Tglad]

Thanks jehovajah, do you mean the dynamic fractals here- https://sites.google.com/site/tomloweprojects/scale-symmetry/dynamic-fractals
I fixed up the bad links, can't seem to link to google drive images even though they were public.

More surprising results, these look strangely 3d given that they are only 2d-

[jehovajah]

Yep! They are the ones.

I keep switching between 2 views! One where the light material is the body, and the other where the dark material is the body. This could explain the 3d quality of the image, because the image processing in our brains is switching between 2 views.

[Tglad]

Changeable weather this evening... stormy-



evening clouds...

[Tglad]

There's a sub family where shifts is zero. Not as much variety, but it appears to be always connected:

[jehovajah]

At what scale are these,Tglad?
Are we looking at part of the traditional Mandelbrot?
If not, what scale do you have to get to to get an analogue?
I notice in the zero c ones some echo of it, but of course you have to be precise if you want to get close to it.
Now I am not suggesting that should be your goal, but from my own perspective I know the Mandelbrot form is very sensitive to coefficient sequence. ,thus  the Mandelbrot  formula actually is evident in many natural growth structures if you take that sensitivity into account.
Nature, unlike our computers, does not follow a rigid "for loop" structure for its iterations!

[Tglad]

The images are -0.5 to 0.5, but the last two are -4 to 4 I think. The last two aren't zero c, there is still c, but the mobius transforms m₂(v) have no translation.
No it isn't part of the traditional mandelbrot, there is no z^2.

[kram1032]

Since these are based on Möbius-transforms, it should be fairly easy to extend them to a "proper" 3D-version, right?
I'd be keen to see how that might look like.

[jehovajah]

No it isn't part of the traditional mandelbrot, there is no z^2.

This is not what I meant by traditional. Rather I am referring to the iconic form. Is it possible to approach to the iconic form with a suitable selection of the transforms? At the moment you have them on a randomiser of sorts I think you said initially. Can you control the rotation and scaling to approach to the iconic form?

[Tglad]

Kram1032- Yes that's right, it should work in 3d, though I think you would want the function to be triple-valued m₃(v) rather than double-valued.
It would be a bit of work though, because tripling each iteration will make rendering slow, and there might need to be a volumetric fog type of render, to render the density value per voxel.

Jehovajah- you'll have to tell me what you mean by 'iconic form' for the Mandelbrot set.

[Tglad]

Here's an example with simple symmetric parameters, which could easily extend to 3d.


shifts is 0, bends are (1,0) and (0,1), and flips are (-1,1) and (1,-1). Giving simpler code:

Code:

getPixelCount(int &count, Vector2 &point, int depth)
  for (i = 0; i<2; i++)
    pos = point - bends[i];
    pos /= dot(pos, pos);
    pos += bends[i];
    pos -= 2.0 * flips[i] * pos.dot(flips[i]) / dot(flips[i], flips[i]);
    pos *= scale;
    pos += C;
    if (dot(pos, pos) > 4)
      continue;
    if (depth == 0)
      count++;
    else
      getPixelCount(count, pos, depth - 1);

[kram1032]

Really nice
With a bit of gradient work, those bulb-like structures could turn out very much like some cartoony fractal flames
You are always showing small sections of the whole thing. Are these structures finite? Could you zoom out to an extend where you can see the whole thing?

[Tglad]

Hard to know actually, the difficulty is defining the escaped region. In Mandelbrot set the escaped points go to infinity and the region is any magnitude > 2, but for these fractals it isn't so simple, and the region changes with C. I don't think the threshold that works for the zoomed in region works on the zoomed out areas, consequently the fractal shapes give way to smoother shapes further from the centre of the image.
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=12425

[cbuchner1]

I think we need a realtime GPU version of this

[kram1032]

good luck with realtime, if the calculation is indeed borderline exponential, heh
But faster should be doable.

I'd love to know how to even find proper escape-domains mathematically.
Like for the m-set, it's not just empirically obvious from experimentally rendering that nothing stays inside the set beyond r=2, it's proven mathematically, right?
If you knew how such things can be proven, maybe you could find the "correct" escape-radius, if it exists at all.