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Author Topic: Quantum Fractal Generator  (Read 31615 times)
Description: Online interactive quantum fractals (based on Mobius transformations)
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Tglad
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« Reply #15 on: September 10, 2013, 12:43:49 PM »

Well people seem to apply Indra's pearls (or some similar Kleinian limit set) to spheres:


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.
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arki
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« Reply #16 on: September 10, 2013, 04:06:03 PM »

Well people seem to apply Indra's pearls (or some similar Kleinian limit set) to spheres:
<Quoted Image Removed>
<Quoted Image Removed>
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 ....
« Last Edit: September 10, 2013, 04:07:39 PM by arki » Logged
Roquen
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« Reply #17 on: September 10, 2013, 04:16:29 PM »

... since these are the conformal set ... required for the result to be fractal and not a stretched out shape.
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.
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arki
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« Reply #18 on: September 10, 2013, 04:45:47 PM »

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.
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arki
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« Reply #19 on: September 10, 2013, 08:18:32 PM »

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

 

Not yet implemented in the generator (3d rendering in java script is still buggy).
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Tglad
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« Reply #20 on: September 11, 2013, 11:45:53 AM »

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

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.

Quote
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.
That's forums for you smiley we're all learning here.
« Last Edit: September 11, 2013, 11:48:58 AM by Tglad » Logged
arki
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« Reply #21 on: September 11, 2013, 12:52:59 PM »

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

Interesting. Where did you get it from?

Quote
Conformal transformations by definition introduce no stretch locally (on infinitesimal shapes).

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.
« Last Edit: September 11, 2013, 02:46:09 PM by arki » Logged
Tglad
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« Reply #22 on: September 12, 2013, 11:58:23 AM »

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 ..

the one I guessed at is similar but has 20 large holes
« Last Edit: September 12, 2013, 12:00:41 PM by Tglad » Logged
arki
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« Reply #23 on: September 12, 2013, 01:41:08 PM »

"Conformal maps preserve both angles and the shapes of infinitesimally small figures"

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.
« Last Edit: September 12, 2013, 02:18:50 PM by arki » Logged
Tglad
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« Reply #24 on: September 13, 2013, 12:40:02 AM »

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.
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arki
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« Reply #25 on: September 13, 2013, 09:12:21 AM »

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" 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."
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Roquen
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« Reply #26 on: September 15, 2013, 04:49:56 PM »

That's forums for you smiley we're all learning here.
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?
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Tglad
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« Reply #27 on: September 16, 2013, 10:56:49 AM »

Quote
Is it provable that if a transform is non-conformal [..] you cannot have detail at all scales as a side effect? Or [..] it's impossible for self-similar structures to appear?
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.

Quote
Is it implicitly assumed that (in 2D cases) we're only talking about systems equivalent to complex numbers?
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)
« Last Edit: September 16, 2013, 11:53:25 AM by Tglad » Logged
arki
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« Reply #28 on: September 16, 2013, 02:22:02 PM »

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:

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). In two dimensions the concept of a conformal map 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.





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?
« Last Edit: September 16, 2013, 04:14:43 PM by arki » Logged
Tglad
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« Reply #29 on: September 17, 2013, 08:48:28 AM »

Quote
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?
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.
« Last Edit: September 17, 2013, 10:14:01 AM by Tglad » Logged
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