Towards a general recipe to explore new fractals?

[knighty]

Many fractals are found/designed by members of this forum, especially 3d escape time fractals. I've noticed that most (all?) of them involve folding and stretching operations. After some investigation, I realized that these concepts are actually used in chaos theory. Nothing surprising, but since Tglad's amazing box, Buddhi applyed box folding to other fractals, David Makin allowed box folding to be applyed a certain number of times as a prepass and last but not least, Jessy allowed to combine fractals... There must be something underneath...

Generalization proposition:
Say we have a set of transformation (mappings) that involve both stretch and fold. Let's calls those transformations T1,T2..Tn. Now let have a choice (or production) function P(k) that returns one of the Ti transformations when given an integer k. Now let the following be our iteration:
z:initial point;
for(i=0;i<max_iterations and somenorm(z)<bailout;i++){//
    transformation T=P(i);//choose transformation
    z=T(z);//apply transformation
}

P() can be a trivial function that always return the same transformation, be cyclic or based on an L-system...
This opens a wide range of possibilities...

[David Makin]

Basically that's what I'm working on now

[msltoe]

The key is to get T(z) to be close to conformal. Otherwise you end up with stretching.
Conformal mapping in 3-D is very limited: scaling, translation, reflection, rotation, and sphere inversion. Tglad's Mandelbox uses all of these but rotations. Also, he made sure the transformations were continuous.
You can derive Sierpinski-like fractals with discontinuous reflections: http://www.fractalforums.com/3d-fractal-generation/sierpinski-like-fractals-using-an-iterative-function/
-mike

[reesej2]

I think your general strategy certainly makes sense but I think we can go further by specifying what the choices of Tn should be. As msltoe said, the transformations should generally be conformal mappings. Also, I've noticed that the best results are usually caused by a conflict of coordinate systems--the Mandelbrot, for example, is largely the product of scaling in polar coordinates (z^2) and translating in rectilinear coordinates (+ c). By contrast, using JUST polar scaling or translation does not produce anything very interesting. So may I suggest this:

z: initial point;
for (i = 0; i < max_iterations and somenorm(z) < bailout; i++) {
    transformation T = P(i);
    coordinate_system C = Q(i);
    z = T(z, C);
}

where T is some conformal transformation, but applied within the context of the given coordinate system.

[David Makin]

Personally I'm not too worried about whether the *individual* transformations applied are conformal or not, once you start allowing multiple transformations of different types per iteration then interesting things start happening and stretching/whipped cream issues can be adjusted by adding more tranforms
Also sometimes whipped cream can be attractive, my best example being this one which is a Julibrot (the classic whipped cream method):



If no image above just click the link.

[Rathinagiri]

Just pleasing. Great. Thanks for sharing

[bib]

Nice shapes and shadows, but I'm not fan of the 3D fbm, although I understand it's an easy way to bring some variety in colors.

[Buddhi]

Beautiful!

[knighty]

Basically that's what I'm working on now

I'm not surprised. I had the feeling that the idea was around. That's why I started this thread.

The key is to get T(z) to be close to conformal. Otherwise you end up with stretching.
Conformal mapping in 3-D is very limited: scaling, translation, reflection, rotation, and sphere inversion. Tglad's Mandelbox uses all of these but rotations. Also, he made sure the transformations were continuous.

I agree with David Makin about this point. For example, z[n+1]=z[n]^p+c is non conformal (I don't know if it is in 2d but in highier dimensions it's not). I think that folding and stretching are fundamental. Some other properties are perhaps necessay. For example the asymptotic behaviour of the whole formula. In the mandelbox or escape time ifs it's z*scale^i, in mandelbrot like fractals it's z^(power^i).

You can derive Sierpinski-like fractals with discontinuous reflections: http://www.fractalforums.com/3d-fractal-generation/sierpinski-like-fractals-using-an-iterative-function/
-mike

This (along with this thread:http://www.fractalforums.com/3d-fractal-generation/tglad's-mandelbox-and-using-the-delta-de-methods-for-rifs/) was the stating point of what i've called "Kaleidoscopic IFS". The key difference is the explicit folding. At first I wanted to make your algorithm GPU friendly.

I think your general strategy certainly makes sense but I think we can go further by specifying what the choices of Tn should be. As msltoe said, the transformations should generally be conformal mappings. Also, I've noticed that the best results are usually caused by a conflict of coordinate systems--the Mandelbrot, for example, is largely the product of scaling in polar coordinates (z^2) and translating in rectilinear coordinates (+ c). By contrast, using JUST polar scaling or translation does not produce anything very interesting. So may I suggest this:

z: initial point;
for (i = 0; i < max_iterations and somenorm(z) < bailout; i++) {
    transformation T = P(i);
    coordinate_system C = Q(i);
    z = T(z, C);
}

where T is some conformal transformation, but applied within the context of the given coordinate system.

I think that your suggestion is equivalent to mine: the coordinate system may be integrated into the transform.
You asked a good question about what the choices of Tn should be.
Other questions are about how to find a good distance estimate (http://www.fractalforums.com/3d-fractal-generation/de-calculation-for-'compound'-fractals/) and smooth iteration.

[reesej2]

Oh, yes, the two ideas are certainly equivalent. I'm just trying to make the properties of the transforms more explicit. This way we can restrict the transforms to conformal mappings without losing stuff like the time-honored z^2 + c, but keep the other properties of conformal mappings. z^p isn't conformal, unless you consider it within the context of a polar coordinate system--if you "unwrap" the plane, you'll get a conformal mapping.

[knighty]

I begin now to understand the relevance of your proposal.

[David Makin]

Oh, yes, the two ideas are certainly equivalent. I'm just trying to make the properties of the transforms more explicit. This way we can restrict the transforms to conformal mappings without losing stuff like the time-honored z^2 + c, but keep the other properties of conformal mappings. z^p isn't conformal, unless you consider it within the context of a polar coordinate system--if you "unwrap" the plane, you'll get a conformal mapping.

Hmmm... I still don't completely follow, since at some point you'll have to convert to cartesian and therefore lose the conformal property anyway - unless I'm missing something that's absent in my maths education, which I grant you is limited to UK "A" level from the early 80's in this context

[reesej2]

You lose the conformal property, but it still has certain conformal-ish properties. I realize that's a fuzzy way of putting things, but hey. It would get rid of the "whipped-cream" effect for the same reason that the Mandelbrot doesn't have whipped cream. I think.