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Buffalo IFS

(http://nocache-nocookies.digitalgott.com/gallery/16/8851_09_11_14_11_36_52.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16758

Once in a while I stumble over really nice locations in the abs-fractals given by stardust4ever.
The ones I find most appealing are those that resembles the classic IFS fractals.

However I have unfortunately a hard time finding these deliberately. I just stumble my way around without any sense of where to find interesting stuff. I consider myself having a decent sense of how to find things in the classic Mandelbrot set, so I guess I only need more practice.

I think it would be awesome if anyone would like to contribute with interesting locations, especially IFS resembling locations, but anything is welcome.

Here is my first contribution, with parameters.

I copied the parameters from the krf file, which is just a text file. The Rotate and Ratio parameters are useful when reproducing such images.

Since I saw the movie below, I feel that what I can come up with is so far from what is possible, so at the moment I kind of lost inspiration on doing more movies:
https://www.youtube.com/watch?v=Diiz1A_NkNM

Very nice. These IFS-shapes are juliasets in some cases. If I want to know where to find which juliaset I mostly use a different fractal application to search for the juliaset I want to use. Did you try this?

Which other application do you use for this?

There are a lot applications with it but I mostly use ''fractal zoomer'':
http://sourceforge.net/projects/fractalzoomer/

Can it be used to find nice juliasets in the abs-fractals?

Yes, it is possible.

Find the ''User Formula''. Open it and import the mathematical formula you want to use following the rules of the application.

At what point do escape time and IFS fractal really differ? I see more similarities, and can hardly explain the big difference. Both have a iterative method, use (a subset of) translation, scaling, rotation, reflection.

The wikipedia formal definition for IFS systems:

Source: https://en.wikipedia.org/wiki/Iterated_function_system

Not really helping me yet ... definition of "contraction mapping" :

Source: https://en.wikipedia.org/wiki/Contraction_mapping

I think you need a shape to start with for normal IFS systems. For escape time fractal, the shape consists of all the points of the real/imag plane (bounded by the calculation window). Maybe the most important distinction is throwing away points that move to far from the begin point. This means we keep all the points that move inwards. This last thing is also true for most IFS fractals. Maybe the points flying away still exists, but they don't really count anyway. You would not see them.

contracting mapping - by contracting mapping is basically meant to have a shrinking procedure, which is normally caused by multiplication factors below 1 ( 1= unity, >1 expanding ) a contracting mapping of a finite transformation can be checked by taking the eigenvalue of the transformation matrix if that is smaller 1 it is contractive

so, for flame fractals more complex transformations are introduced the contractive behaviour is needed because if nothing would contract you would never see anything on your screen because everything blows away, so the flame fractals contain both transforms some expanding, some contracting and some unit transformations which dont change the absolute distance. in combination the propability for the contractions need the be high enough to bring stuff back closer to zero by combining such the beautiful fractals we all like so much is created



formally both are iterated function systems, for ifs systems as you know from apophysis they are propabilistic iterated functions, which means any function has a certain propability to be choosen from

the difference of escape time and flame/ifs is as follows:

in escape time fractals you examine each dot separately, perform a number of iterations and check if it is inside or outside the predetermined bailout value - check my mandelbrot videos for a visualisation of the process

for so called ifs/flame fractals the process is to take a single starting point - in fact it does not even matter what the starting point is, because the result is the strange attractor formed by the transformations, and the starting point will always approach this limit - and color the flame fractal by counting how often this point hits the pixel position on screen, the iteration count is very huge to obtain an image colored in this way....

All IFS fractals can be formulated as an escape-time fractal. The KIFS fractals are the most obvious example, but it's even possible with trickier ones such as the Barnsley Fern. In the fern's case, an iterated point will often branch into several points, and keep branching, and you have to essentially traverse a tree and make sure none of the iterates stay bounded. But I would still call it an escape-time fractal.

Anyway, the fundamental difference between traditional escape-time fractals and IFS is that the iteration function for escape-time fractals is always continuous(but not always differentiable), whereas with almost all IFS fractals, the iteration function is discontinuous, or "broken", as a different function is applied based on the location of the point. I think even KIFS fractals have discontinuous functions.

Ok so all IFS fractals are there, somewhere.
Where...?

Uh... That's a simple one.
IFS are (a subset of?) the space of all systems of maps of systems of maps to systems of maps.
Something like that should, I'm pretty sure, be neatly formulable in some adequate kind of foundational theory.

The question "where" they are is much less interesting than "which ones" actually are interesting (for some suitable definition of "interesting" which most likely will be something like "of interest to your personal perception")

You mean in the Buffalo fractal? No, I don't think every IFS fractal can be found within an abs() mandelbrot variant. I don't know if you could even find something simple like the Sierpinski triangle. I've seen similar shapes within the interior of the Burning Ship, but nothing exactly like it.

Yes, that is what I meant.
The movie from fractal universe contains both Sierpinski triangle, Koch snowflake and others, but unfortunately he didn't kept the locations.
I would like a catalog of nice locations, because I know they are there but I still think they are hard to find.
But I will start finding them :)

so, to open up your mind a little, the koch/sierpinski structures are similar to what is found in the mandelbox, lets discuss a little more openly, and think about the relationship of the formulas, they all share the folding ... the folding is parametrized in the mandelbox, the abs function fold at the axis... keep it coming what is about the folding that leads to the sierpinskis ??? ;)

I managed to find a sierpinski-like shape in the cubic buffalo:
(http://i.imgur.com/ogy98Vv.jpg)