New type of fractal

[Tglad]

At least I think it is new 
I'm combining the idea of a kleinian fractal with the +c part of the mandelbrot set. So I guess this is making a Mandelbrot set of Kleinians.
I have two transforms each iteration. Each transform is a combination of a translation, scale, rotation and invert-reflect. The 4 possible conformal transforms.
I then add c, and colour each pixel based on how many paths stay within the escape radius.
Here are some pics for different transforms. Notice that each pic has variation within the fractal shape-

[kram1032]

you like your red cols, don't you?
Certainly looks like nice stuff. Shapes remind me of both mandelbrot-sections and shape-based IFS patterns.

[Tglad]

some more, with a temperature colouring:

[Tglad]

and more

[kram1032]

that brings out the details a lot better

[KRAFTWERK]

Mysterious stuff...  
I will follow the progress!

[jehovajah]

that brings out the details a lot better

It sure does.

I am interested in the programming structure here. Presumably you use procedural calls in place of the usual z notation, but the c is your seed vector. What is your vector space complex plane, quaternion block, or just simple Cartesian vector?

Does it matter which you choose?

[Tglad]

Hi Jehovajah, it uses 2d vector algebra, replaces:
 z = z²+c
with
 v = m₂(v)+c
where m₂(v) is a double valued function, giving two different mobius transforms of v. A mobius transform is either a translation, rotation, invert-flip or scale, or a combination of any of these. In the pictures I'm using a random combination, and scaling by 2.
It would work in more dimensions, but you might need to triple each iteration or more, and it would be slow because of all the paths to check.

Very stormy look here:

[hobold]

giving two different mobius transforms of v

So the number of orbits that you are tracking doubles with each iteration?

[Tglad]

Yes, but if a point goes outside a threshold then you terminate that route and its children, so in practice you aren't necessarily doing 2^n checks for an iteration depth of in.
Here's another-

[jehovajah]

Some great effects and very interesting because it is just simple vector "dust".

Using your approach I feel you can more readily model natural physical forces in interpretable ways. There is always this suspicion about the imaginaries, which I have satisfied myself is totally unjustified, because they are not "imaginaries" but constant ratios of semi circular arc to diameter.

Your method allows one to think say of a plasma acting on the atmospheric water molecules.
The plasma drives rotates and flips water molecules about by electrostatic and magnetic action. The scale factor allows us to zoom in and see particular patterns of droplets at a given instant.

Of course this is a simplistic description, but who wants to make it hard! Well, maybe a research physicist might, but I think the principle is simple enough.

[Tglad]

Its fun exploring images from this fractal...









I haven't got the right threshold value working yet, so there are some artefacts, like hard lines... but in this pic I think it makes it look quite artistic-


Probably not the best formulation yet, but here's the code so far:

Code:

recurse(int &count, Vector3 &point, Vector3 &C, int depth) 
  for (i = 0; i<2; i++)
    // 1. this is a invert  (which naturally flips the handedness)
    Vector3 pos = point - bends[i];
    pos *= bends[i].magnitudeSquared() / pos.magnitudeSquared(); 
    pos += bends[i];
    // 2. so a (second) flip is necessary (so not anti-conformal)
    pos -= 2.0 * flips[i] * pos.dot(flips[i]); // flips is normalised, 2 flips make a rotation
    // 3. arbitrary translations
    pos += shifts[i];
    // 4. scale up
    pos *= scale;
    // 5. add the offset (the initial pixel position)
    pos += C;
    if (pos.magnitudeSquared() > threshold) 
      continue;
    if (depth == 0)
      count++;
    else
      recurse(count, pos, offset, depth - 1);

In my tests:
bends is a random -5 to 5 vector, flips is a random normalised vector, shifts is random -0.5 to 0.5 vector.
threshold is 9, scale is 2 and depth starts at 16. I am viewing the fractal in a window between -0.5 and 0.5.

[taurus]

the images quit loading on my 'puter with an error 403 - forbidden! i guess i am not the only one 

[Tglad]

I added the images here too- https://sites.google.com/site/tomloweprojects/scale-symmetry/mobius-maps

[taurus]

thanks, this site loads properly.

the ones in the previous post seem to be placed in a folder within your google account, that prohibits public acces. most likely you are the only one, who sees them.

my understanding of what you ar doing here is very limited, but it's still interresting to see how new ideas grow.
visually they remind me more of outside julia sets - they look some sort of fragmented.