Quantum Fractal Generator

[arki]

Available online version of Quantum Fractal Generator. Still preliminary version. New options will be added. Documentation will grow (it is minimal right now). Bugs will be fixed. But it already allows you to generate a whole family of quantum fractals, also in high resolution, using either chaos game algorithm or (new) Frobenius-Perron deterministic algorithm.

The source code is available on github.

Comments and questions are welcome.

Example:

[bib]

Cool!

The similarity with Abox-based pseudo-kleinan in Mandelbulb3D is striking.

[arki]

Yes, indeed, the similarity is striking. Thanks. There must be some mathematical reason for  it. To be discovered.

[Nahee_Enterprises]

    Available online version of Quantum Fractal Generator.
    Still preliminary version.  New options will be added.  Documentation will grow (it is minimal right now).  Bugs will be fixed.
    But it already allows you to generate a whole family of quantum fractals, also in high resolution, using either
    chaos game algorithm or (new) Frobenius-Perron deterministic algorithm.
    The source code is available on github.
    Comments and questions are welcome.

Gave it a quick try, going through the different Types with various settings.  Did notice that there does not seem to be much of a change when using either the "Rain" or "Spectrum" color schemes, they appear to be quite similar.

This is an interesting beginning of your online application, and will be awaiting further enhancement announcements.     
 

[arki]

Did notice that there does not seem to be much of a change when using either the "Rain" or "Spectrum" color schemes, they appear to be quite similar.

You are right. Thanks. In fact, some color schemes look good for this kind of fractals, some tend to stress inessential features and do not look cool at all. The options will change after some experimenting.

[eiffie]

Very interesting, thanks for the links!

[Tglad]

Interesting links with special relativity and quantum theory but these fractals seem to already exist under many names such as Indra's pearls, IFS, pseudo-kleinians. Since most fractal generators work on a repeated set of mobius transformations, since these are the conformal set (which also exist in more dimensions) required for the result to be fractal and not a stretched out shape.
Is there a suggestion that some real world quantum wave functions would be fractal like these?

[arki]

Interesting links with special relativity and quantum theory but these fractals seem to already exist under many names such as Indra's pearls, IFS, pseudo-kleinians.

IFS is a generic name for "iterated function system". Indra pearls are different. They live on the Poincare disc (non-compact space), not on the sphere that is compact. Etc. etc. One may say: all fractals are similar. And yet there are many different families of fractals with different properties. The devil is in the details.
For instance, according to a recent paper by Andrew Vince et al. on Mobius IFS, the last fractal ("para") in the Generator has no rights to exist. Vince proved (with Barnsley) that parabolic transformations can not generate fractals. And yet we have something that looks like a fractal and walks like a fractal. It is no yet clear if it quacks like a fractal. Vince himself was puzzled when saw this fractal

Since most fractal generators work on a repeated set of mobius transformations, since these are the conformal set (which also exist in more dimensions) required for the result to be fractal and not a stretched out shape.

Those on the sphere are partially stretched. They are not made of contractions.

Is there a suggestion that some real world quantum wave functions would be fractal like these?

It is not that simple. The distribution (measure) on the sphere represents a density matrix (mixed state), not a wave function. Moreover, because of linearity of quantum mechanics many different distributions represent the same mixed state. So, according to the orthodox interpretations of quantum theory these fractal measures will not be observable. Yet there are also non-orthodox interpretations (you have to get rid of the orthodox interpretation of the Heisenberg's uncertainty relations first) that suggest that these fractal shapes can leave experimental traces. So there may be some new adventure ahead...

P.S. And indeed, there are fractals made of Mobius transformations living in higher dimensions. Especially nice are those in 4 dimensions, since in 4D we have beautiful Platonic solids (24 cell and 600 cell). They will be added to the generator (3D slices) in due time.

[lycium]

Thanks for your interesting posts, and welcome to FractalForums! I'm always excited to see more knowledgeable IFS people here

All the best with your experiments.

[Tglad]

Yes, most fractals are either infinite in their size or live on the Poincare disc, but both can be easily transformed onto a sphere conformally. Circles remain as circles.
Anyway, its an interesting slant, from a physics angle, I'll have to have a go with the 'para' fractal.

You might be interested in the Kleinian spheres section of this page: http://www.aleph.se/andart/archives/2005/11/

[arki]

Yes, most fractals are either infinite in their size or live on the Poincare disc, but both can be easily transformed onto a sphere conformally. Circles remain as circles.
Anyway, its an interesting slant, from a physics angle, I'll have to have a go with the 'para' fractal.

You might be interested in the Kleinian spheres section of this page: http://www.aleph.se/andart/archives/2005/11/

Very interesting. It seems you have discovered another way of generating Platonic quantum fractals, in particular the octahedron:

http://xxx.lanl.gov/abs/quant-ph/0204056 (in more than 3D: http://xxx.lanl.gov/abs/quant-ph/0608117 )

Food for the thought.

By the way: what is your formula for transforming conformally the Poincare disk into the sphere?

[Tglad]

A few more Kleinian fractals here: http://www.josleys.com/show_gallery.php?galid=346  3d but not on a sphere surface.

By the way: what is your formula for transforming conformally the Poincare disk into the sphere?

Any conformal fractal on the 2d plane converts to a Reimann sphere using the normal formula. So a conformal fractal on the poincare disk can go straight onto a sphere, but it'll have a big gap from magnitude 1 to infinity, which you can close just by enlarging the disk... or apply other mobius transforms if you wish.

The quantum take on it is interesting, I wish I had time to read more about it.

Straying off topic, but you might also like these which I made, they bear some resemblance also, to your fractals: https://sites.google.com/site/tomloweprojects/scale-symmetry/mobius-maps

[arki]

What is the "normal formula" for a conformal map of the Poincare disk into the sphere?

[Tglad]

Any image (including Indra's pearls) which exists on a disk can be mapped conformally to a sphere surface. The disk is on a plane, and each point is identified by the polar coordinates (distance, angle), the equivalent sphere is given in spherical coordinates where longitude = angle and latitude = 2*atan(distance) - 90. You can scale the disk on the plane by any amount prior to converting to a sphere, so the 'hole' at the sphere head can be as small as you like. The image on the sphere is then conformal, infinitesimal shapes are preserved.

[arki]

Any image (including Indra's pearls) which exists on a disk can be mapped conformally to a sphere surface

This is "cheating". This is painting the disk on the sphere. But action of Mobius transformations on the sphere and action of Mobius transformations on the painted disk are not the same. They simply can't be the same since it is PSL(2,C) that acts on the sphere and it is PSL(2,R) that acts on the disk. The disk and the sphere have different groups that acts on them.