Welcome to Fractal Forums

Available online version of Quantum Fractal Generator (http://arkadiusz-jadczyk.eu/blog/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 (https://github.com/jasonknight/jfractals).

Comments and questions are welcome.

Example:

(http://arkadiusz-jadczyk.eu/images/para1.jpg)

Cool!

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

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

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.    :D 
 

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.

Very interesting, thanks for the links!

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?

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


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


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.

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.

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?

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

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

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

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.

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.

Well people seem to apply Indra's pearls (or some similar Kleinian limit set) to spheres:
(http://page.math.tu-berlin.de/~gunn/Pictures/PuncturedTorus-400.jpg)
(http://i.imgur.com/3jzgU.jpg)
This might be interesting too, as it is similar: http://www.fractalforums.com/other-types/kleinian-limit-sets-t2270/

Anyway, regardless, it is good to compare with similar results, to see where the differences are.

Thanks. Do you know how exactly was this last image obtained? I could not find the description of the algorithm. I would like to understand the similarities and the differences. Certainly this is not a transcription of Indra pearls from the disk to the sphere! Perhaps there is something that can be learned ....

I'd appreciate a citation for this.  I keep seeing it repeated (here and elsewhere) but have had zero luck finding any proof or even basis for the statement.  Moreover I've seen unquestionably incorrect mathematical statements made by some of the people repeating this so I'm very skeptical about it's truth.

Indeed. As for the family of Mobius IFSs I am interested in, I can prove (for a while by numerical estimation, I do not have an analytical proof yet), that they are "contractive on average". And the proof is not that easy.

Here is 3D rendering of the "parabolic fractal" with Frobenius-Perron algorithm (level 3) - stereographic projection from the sphere:

(http://arkadiusz-jadczyk.eu/images/para_fp_stereo_level3.jpg) 

Not yet implemented in the generator (3d rendering in java script is still buggy).

Arki, I don't know the exact formula for the large image but it looks like an icosahedral sphere inversion limit set.

Conformal transformations by definition introduce no stretch locally (on infinitesimal shapes). More importantly non-conformal transformations by definition do introduce local stretch. You can google to verify these facts.
Fractals are produced by repeated transforms so several people on the forums have noted that a later transform could 'unstretch' the stretching effects of a non-conformal earlier transform. This is true, but such a system is equivalent to the application of a set of only conformal transforms; the final combined set of transforms is itself a transform, which must be conformal to produce non-stretching fractals.

It isn't clear that transformations are always needed to create fractals, e.g. diffuse limited aggregation, but there is an implicit conformality in the use of the Euclidean distance function in this case I think.

I suppose it is wrong to say 'conformal transforms are required for it to be a fractal' since several 'fractals' are stretched, like the mandelbulb, trihorn, burning ship. But I'm not sure to what extent these are fractals in the strictest sense... they are not self-similar in shape... it depends on your choice of definition of the word fractal.. there is some leeway.

That's forums for you :) we're all learning here.

Interesting. Where did you get it from?


I am curious: Where did you get this "definition" from?

I am sensing some misunderstanding here, and I would be happy to explain some concepts, but first I would like to know the source of this "not quite exact "definition". Perhaps it would be useful to discuss this subject in some detail, as apparently there are others that would like to see the mathematical statements correct.

I wouldn't call it a definition. By definition a rectangle has straight edges but it doesn't follow that this is my definition for a rectangle. However I can verify that fact about conformal transforms:
"Conformal maps preserve both angles and the shapes of infinitesimally small figures" Wikipedia, http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Conformal_map.html
"Conformal transformations, or mappings, have many important properties and uses. One property relevant to image transformation is the preservation of local shape" http://www.mathworks.com.au/help/images/examples/exploring-a-conformal-mapping.html

It is more commonly written that conformal maps are locally angle preserving, the preservation of infinitesimal shapes is a direct result of this.
"A conformal mapping, also called a conformal map, conformal transformation [...] is a transformation w=f(z) that preserves local angles. ..." Mathworld.wolfram.com

Arki, It looks like Sandberg's (previous link) Kleinian spheres, Sandberg shows a cubic version and an octahedral version ..
(http://www.aleph.se/andart/archives/images/zoct1.250000.png)
the one I guessed at is similar but has 20 large holes

That is fine. But saying that they are not "stretching out shape" is something different. First of all "stretching depends on the metric (how you measure distances). On the Poincare disk, for instance, there is a non-Euclidean meatric and there is the Euclidean metric (inherited form the plane on which the disk is being placed). They are different. Conformal transformations there preserve non-Euclidean distances, but do not preserve the Euclidean ones. On the sphere there is no conformally invariant metric. When we take the natural distance on the sphere (along great circles) - it is not preserved by conformal transformations (except for pure rotations). Conformal transformations will stretch or contract, even infinitesimally  (depending on the point). The same holds on the plane, if you project the sphere stereographically (minus one point). The Euclidean metric on the plane is also not preserved, conformal transformations stretch or contract, but circles go into circles (or straight lines).

Conformal transformations on the plane form a 6-parameter group. Essentially SL(2,C) - the group of 2x2 complex matrices of determinant 1. If the plane is thought of as the complex plane: z=x+iy, then conformal transformations act as
z -> (az+b)/(cz+d)
Among these transformations there are special ones, belonging to the 3-parameter subgroup SU(1,1). These transformations map the unit disk into itself. So, the Poincare disk can be thought of as being part of the plane (or on the sphere) provided you restrict your conformal transformations to the smaller group (essentially the same as SL(2,R)). This smaller group is being used in Indra's pearls. I am using the larger group. Sanderg evidently also used the larger group (SL(2,C) or SO(3,1)). As he is not revealing his algorithm, it is possible that has used the one described in my 2002 paper.

By a stretched image I mean one that gets longer in one axis than another, I don't mean dilation/contraction which are of course allowed in a conformal transform.
OK lets avoid the ambiguous word stretch and agree that conformal transforms are locally shape preserving.

That is better, but also not precise enough. We must first make precise on which space conformal transformations are acting and how we define the distance (metric) on this space. Quoting from the paper "Generalized Fractal Transforms and Self-Similar Objects in Cone Metric Spaces (http://math.acadiau.ca/mendivil/Papers/ConeMetrics.pdf)" by H. Kunze et al:

"Properties of contractive mappings are used throughout mathematics, usually to invoke Banach's theorem on fixed points of contractions. In the classical case
of iterated function systems (IFSs), the existence of self-similar objects relies on this same theorem. Fundamental ingredients of the theory of IFS are the use of
complete metric spaces and the notion of contractivity, which both depend on the denition of the underlying distance."

I don't want anyone to misinterpret my statement.  Everyone on the planet is a math n00b...it's the nature of the beast.  Incorrect or unfounded statements outside of formal writing (esp on fourms and blogs) is to be expected...I brain dump garbage frequently.  So no disrespect was intended.  The reason why I bring this up is that it seems to be accepted and yet my mathematical knowledge doesn't allow me to connect the dots.  So I question if it is really true or has just become the accepted "truth" due to repetition.

It's been my assumption that the reason conformal mapping is considered to be a requirement is the effect it has on the domain, I'll vulgarize and say distortions and/or orientation reversal.  Is it provable that if a transform is non-conformal there there cannot exist a connect path through the domain which is infinitely transformed or more simply you cannot have detail at all scales as a side effect?  That seems unlikely to me.  Or is it provable that somehow it's impossible for self-similar structures to appear?   Is it implicitly assumed that (in 2D cases) we're only talking about systems equivalent to complex numbers?

I don't know a specific proof (do you Arki?), but less formally it makes a lot of sense and is a good bet based on results:

Conformal:
Almost all the nearly 90 deterministic fractals here: http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension including all KIFS too
(The 2 attractors may not be conformal, but they probably aren't exactly self-similar either)

non-conformal:
burning ship, mandelbulb, tricorn  <-- these all look elongated in places

In the case of the mandelbox there is still a lot of detail at all levels despite it being only semi-conformal, so there is still stretch in some places.

You can do anything with complex numbers that you can do with 2d vectors, so the systems are equivalent. Complex numbers are incredibly handy for talking about 2d fractals though because basic operations like z^2 and z^3 and / z etc are all conformal... even sin(z) etc (except some poles)

The devil is in the details. In this case the relevant detail is a precise mathematical definition of self-similarity. If you give me one, then I will try to find the answer.

Moreover, the devil is also in your definition of a conformal map. See Liouville's theorem (conformal mappings) (http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28conformal_mappings%29). In two dimensions the concept of a conformal map (http://en.wikipedia.org/wiki/Conformal_mapping) can be easily confused. Conformal maps there are not the same as Mobius transformations. In higher dimensions (n>2) things are simpler.

Here is an example of an IFS (image rendered through Chaos Game algorithm) consisting of six Mobius transformations with octahedral symmetry, acting on the sphere, then projected stereographically on the plane. The first image is the the central square -8<x,y<8. The second image is x8 zoom, showing the square -1<x,y<1. Certainly it looks self-similar. But, to be fair, I do not yet how to prove that it is "self-similar". I do not even know if it is "a fractal", because there are many different definitions (often fuzzy) of fractals.

(http://arkadiusz-jadczyk.eu/images/oct_st_8.jpg)

(http://arkadiusz-jadczyk.eu/images/oct_st_1.jpg)

We see a pattern of circles. My question is: is there anybody in the world who, just looking at the transformations generating these circles, would be able to tell me what are the formulas for these circles? What are the formulas for their centers and their radii? There must be a formula. But what kind of a formula? Simple algebraic expression? Or rather an algorithm for calculating but not a closed formula?

Well, the 45 degree angle circles should be easy, say you start with circles at (+-1,+-1) with radius 1, then the 4 inner would be (+-a, +-a) and radius a where a is (1-sqrt(0.5))/(1+sqrt(0.5)), this scaling can be repeated to get all the inner (and outer) 45 degree circles.

To get the other circles you can do a mobius transform from the 4 start circles to the 4 slightly smaller ones to the right, it converts the circle of radius sqrt(2)+1 to the vertical line and translates (sqrt(2) + sqrt(0.5)) to (0,0)...
so there is enough information to work out the position of the circles I think... but the formula would be recursive.

First I thought it may be so. But nope. I posted the problem also on physicsforums (http://www.physicsforums.com/showthread.php?t=710747). As you can see there, there is some progress, but still no clear solution.

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))

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

(http://arkadiusz-jadczyk.eu/images/guess6.jpg)

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.

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.

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"?

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

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:

(http://arkadiusz-jadczyk.eu/images/octasphere.jpg)

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]:

(http://arkadiusz-jadczyk.eu/images/kola_g5.gif)

Square [8,8]:

(http://arkadiusz-jadczyk.eu/images/kola_g58.gif)

Another update:

Birth, Life and Death of a Quantum Fractal

http://www.youtube.com/edit?video_id=lRHl27H2UpI