Quantum Fractal Generator

[arki]

Well, the 45 degree angle circles should be easy...

First I thought it may be so. But nope. I posted the problem also on physicsforums. As you can see there, there is some progress, but still no clear solution.

[Tglad]

I'll do it with vector algebra, though it might be simpler with complex numbers:
Vector v = (x,y) on the plane.
Start with one quadrant circle:
 (x - 1)^2 + (y-1)^2 < 1
This gives us all four quadrant circles:
 (x,y) = (y,-x)   // defines 90 degree symettry
Next get all the larger and smaller 'quadrant' circles:
 a = (1-sqrt(0.5)) / (1+sqrt(0.5))
 v = av 

Next get the mobius transform:
 b = sqrt(2) + 1
 v = b^2/(v + (b,0)) - (1,0))

[arki]

This gives us all four quadrant circles

But, you see, the picture is misleading, because for a different value of the parameter in this fractal family we get this:



I have guessed the formula for the white circle (red on the picture) for any value of the parameter, but the black round holes (repulsive regions) are still a mystery to me.

[Tglad]

It isn't too surprising that a different parameter will give different results... whiter ones probably correspond to 'less simple attractors' or transforms that line up less well.
Nice picture by the way.
Incidentally, the circles appear to be the same, just smaller radius.

[arki]

Incidentally, the circles appear to be the same, just smaller radius.

Yeah, but I am not sure if the scaling ratio is the same as with the other parameter or not. In principle I could measure it on the computer screen, but would I know then "for sure"?

[Roquen]

Moving my "hijack" of this thread to here: http://www.fractalforums.com/new-theories-and-research/why-there-isn't-3d-12-conformal-transformation/

[Roquen]

To call my topology weak would be being excessively generous to myself...so my with-a-grain-of-salt random thought is: what does it look like if you flip 'z' before projecting back into the plane.

[arki]

To call my topology weak would be being excessively generous to myself...so my with-a-grain-of-salt random thought is: what does it look like if you flip 'z' before projecting back into the plane.

If this is a question about my particular octahedral fractal - it will look exactly the same. The fractal is based on the octahedron, and octahedron is North-South symmetric:

[arki]

I think I am getting at the formula. At least there is some progress. Here are circles made of circles. There is nothing random. All deterministic. And yet it starts looking like my fractal. The point is that we can make fractals out of circles. This opens a new territory? I don't know.

k=0.6. Square [-1,1]:



Square [8,8]:

[arki]

Another update:

Birth, Life and Death of a Quantum Fractal

<a href="http://www.youtube.com/v/lRHl27H2UpI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/lRHl27H2UpI&rel=1&fs=1&hd=1</a>