Spent all day on julia set iim algorithm.. trying to develop a method to fill in the spirals evenly. It isnt easy because the transform is nonlinear. Went to a recursive deterministic method. There is a critical exponent that I believe is related to the exact fractal dimension. My guess is the exponent has to be correct to 4-5 decimals to get the spirals to fill.
2/15/16
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+Adam Majewski
I couldn't get it right. I don't fully understand it. I really don't like the hit table method.
I know a little about the different starting points.
What I was trying was to traverse the binary tree of choices of inverse maps, in an imbalanced way, so that the image is evenly filled in.
This is easy and works well for affine maps in ifs systems, and produces perfect evenly covered renderings of attractors, even when the maps have different scaleing.
The maps that represent the two roots are nonlinear, and this changes alot. These two maps aren't even contractive everywhere on the attractor. If you choose special sequence of roots, and apply those maps successively to a small starting shape, they can be verry expansive.Feb 23, 2016
+Adam Majewski
Many years ago I used DEM.. It's been a while since I've coded it. I need to refresh on the algorithm. It does produce beautiful pictures.Feb 23, 2016
+Adam Majewski
I've been trying to understand field lines and some other things. Was looking at possibly using a Julia sets transforms for data encryption. Mostly because of the things we know about the mappings on Julia sets, and because the complexity is hard to describe by any method except carrying out the computations, maybe it's hard to decrypt a code resulting from these computations.Feb 23, 2016
+Adam Majewski
I see what you mean, easy case and hard case with the tree depth issues. This is what I'm trying to understand. What is the imbalanced tree depth that makes the image rendered evenly. Maybe I need to have some change in the way I describe the maps. I think you are looking at this same phenomenon. I love your renderings and descriptions of algorithms.
I am curious of this reference where you mention "Iterates of critical point delineate a Siegel disc"..Feb 23, 2016- Connected Julia set consist of components. Here :
https://commons.wikimedia.org/wiki/File:Quadratic_Golden_Mean_Siegel_Disc_IIM_Animated.gif
look at level 0 : forward orbit of critical point aproximates boundary of componnet with Siegel disc.
All other ( smaller) componnets are preimages of this 0 componentFeb 23, 2016 - +Juaquin Anderson
You can use code from https://commons.wikimedia.org/wiki/File:Quadratic_Golden_Mean_Siegel_Disc_IIM_Animated.gif
change only c :
const double Cx=-0.74543;
const double Cy=0.11301;Feb 23, 2016