This images show the attractors of some 2D Iterated Function Systems (IFS). The functions are just linear functions, affine transformation if you want, of the form
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that can also be written as |  |
where the contraction fractor is just the determinant of A (since the transformations are linear)
Once selected four transformations for each image, you can transform the plane iteratively with this functions/mappings. Assuming the overall transformation is contracting, the iteration makes the complete plane converge to a given set, depending on the transformations used. The geometry of this set, the attractor of the IFS, is usually fractal, even if the transformations are linear.
But how to transform the complete plane, let's say, one million times? We can easily get the attractor set using other tricks. According to the chaos game, you can recover that attractor by just following the "orbit" that a given point of the plane creates when transforming it iteratively with this transformations, once at a time.
To become this true, chaos theory and symbolic dynamic mathematics say that you must randomly choose the transformation from the transformation set of the IFS. This way, the orbit densely overlaps the attractor. In the other hand, to allow the chaos game converge faster, it is better not to randomly select a transformation from the transformation set of the IFS, but with a probabilty propotional to the area covered by this transformation over the total area of all the transformations:
To make things more interesting, it is also possible not to only record the attractor, but also the densisty of the attractor. This can be interpreted as the probabilty of the iteration to step at a given point across the orbit. After selecting the probabilities according to the equation above, this density over the attractor should be quite smooth.
Finally, a bit more interesting images can be created if other transformations can be applied before accumulating the orbit on the plane. It is important to note that to keep the chaos game unaltered, this transformations must be done out of the iteration feedback loop. To create this images, a simple rotating transformation was applied, in wich the rotating angle was proportinal to the signed horizontal distance from the point to the origin. Additionaly, a sinousoidal omponent was added to this angle, that changed each iteration. Adjusting the frequency and amplitude of this sinusoidal compnent, the thikness and intensity of the "arcs" on the picture can be adjusted. Without this variation, the arcs would become just "threads".
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