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Hi everyone.

Has anyone ever tried to understand what is going on on our Kaliset fractals ? Why is it a fractal ?
What is really happening in the progression of complex numbers associated with a complex number on the c-plane ?


Here is what I've come up : let f be a complex-differentiable function depending on two complex variable z and c, and g be a complex-differentiable function depending on c.
Let foldingN be the continuous but not differentiable complex function that we use to mirror complex numbers, defined on that post : http://www.fractalforums.com/new-theories-and-research/extended-kaliset-t10372/ (http://www.fractalforums.com/new-theories-and-research/extended-kaliset-t10372/). For example, N=2 gives the conditional conjugate, N=4 gives the classic abs function.
let u be the progression of complex numbers such as : u_0 = g(c) and u_{n+1}=f(foldingN(u_n),c).
I think a lot of our Ducks fractals, if not all of them, can be composed like this (remember that f and g are varying on the c-plane).

Of course we don't have necessarily to use foldingN on the center 0, and with a starting angle of 0, but if we want to change that we just have to compose with an affine complex function z->(az+b) to change the center and the angle of the folding. However this can be put in the function f.


One question is : for n an integer, is the function u_n that depends on the complex variable c a piecewise differentiable function ? (it is continuous).

Every time we use the foldingN function we break again the function into N more pieces... and after n iteration if we trace a line everytime we have broken the original function g(c) we should have a tesselation of the c-plane !
And maybe that our coloring method are revealing theses patterns (and maybe they forget some ones because they are too smooth some times).

That could explain a little the colors of our fractals.
But that doesn't tell us what is really going on in u_n(c)...

I think the proper way to think about these fractals is to see them as defined on the sphere, via the stereographic projection. See this blog post for more about how to see these fractals as living on the sphere:
http://algorithmic-worlds.net/blog/blog.php?Post=20120316
We are then studying the iterations of either conformal (for dense Julia sets) or almost conformal (for Ducks-type fractals) maps of the sphere to itself. By "almost conformal", I mean a map that fails to be conformal on a subspace of dimension 1 on the sphere. This is the case for all the implementations of Ducks-type fractals I know of. The subspaces of dimension 1 are then the axes of the mirrors symmetries. I am not sure if the Ansatz you propose can really account for all these almost conformal maps.

What happens to the orbits is the following. Basically there are two types of orbits. Either a neighborhood of the point is mapped smoothly along the orbit, the point is said to be "regular". Else, the map is chaotic in a neighborhood of the point and the point is "irregular". Irregular points belong to the Julia set, yielding interesting visual structures, while regular points form Fatou domains, that appear visually smooth. For some maps, it happens that the Julia set fills the whole sphere, i.e. all the orbits are chaotic. This is what yields dense fractals. You can check explicitly by plotting the orbits that they do become visually chaotic precisely when the pattern fills the plane.

Iterations of conformal maps have been studied extensively in the math literature, see for example this nice review if you have some background in mathematics:
http://arxiv.org/abs/math.DS/9201272
On the other hand, I haven't been able to find a single reference on the "almost conformal" maps that are responsible for Ducks/Kaliset fractals. This is something much less natural to study from the point of view of mathematicians.

I hope this helps,

Sam

Thanks. I guess I have a lot of reading to do !

The most fascinating and appealing I found "entangled trees". This pattern should be something mathematical as it appears from time to time with different formulas, even not a kaliset formulas. Some mix of parabolas or hyperbolas in complex plane???

(http://fc06.deviantart.net/fs71/i/2012/124/c/2/garden_of_eden_by_edo555-d4yi0rw.jpg)

Havent read jet, I will download and read at home.

Yeah i like them too even if I don't understand WHY they appear !
Hey we have the same kind of forest of entangled trees  ;D :
http://www.youtube.com/watch?v=M1iETmCZIfs (http://www.youtube.com/watch?v=M1iETmCZIfs)

In points where 4 tree branches meets there are some kind of spirals. Lines looks more like hyperbolic function, and not the curved second degree lines of mandelbrot, and in formula z=abs(z)/c+c there are just power 1. 
So these probably must be made by hyperbolic function.

Which formula did you used?

I think, "entangled trees" should be result of mirror symetry applied on the hyperbola y=1/x.
(http://upload.wikimedia.org/wikipedia/commons/thumb/2/29/Rectangular_hyperbola.svg/330px-Rectangular_hyperbola.svg.png)

Curves do not have accelerateing curvature as squared or higher degree, alsou in tree formulas there are no squares or higher powers. Trees seems to be a bitt more curved than regular hyperbola are, but when bitt changing julia seed appears regular pattern of hyperbolas without fractal features.
p.s.
This spot alsou is my contest entry for animations.

I don't understand where you're going.
The fractal, if we stop at a finite iteration, is mathematically a finite progression of complex numbers from a starting complex... What we "see" is up to the coloring method...
It's kinda hard to define what we are talking about when we say "this is like an hyperbola".
Saying "it is an hyperbola" is weird because the coloring method is continuous. WHAt mathematically IS the hyperbola you're talking about ?
But i'm not saying you're wrong! You're maybe on something :) .

Anyway, when I'll have the time I'll post some pictures related to the differentiability of the the complex function associated with the last iteration in Julia mode. I think i'm on something interesting.

Cool

fractus

Many colouring methods make thees trees, they are more result of formula. Basic quadratic formula z=z*z+c don't generates trees.

Trees, I think, are just mirrored halves of hyperbolic curves. It seems that trees are drawn only by formulas having hyperbolas as some sort of solutions. That is having division or negative power what is the same as division. And of coarse magic ABS function.


Here is made of stright lines:
(http://rosettacode.org/mw/images/8/8a/MathFractalTree.png)