Self-similar designs from repeated plane mappings and anamorphosis

I am reconsidering ideas Pickover has presented in his book  “Computers, Patterns, Chaos and Beauty”. My post “Fractal surprise from complex function iteration” discusses already some aspects.

pickoversComplex numbers z=x+iy represent the (x,y)-plane and complex functions f(z) define a mapping of the plane. Here I am using f(z)=z*z+0.7. Following Pickover, we use for each pixel with coordinates (x,y) the corresponding complex number z and repeatedly apply the function f on it. For most pixels the result grows without limit. Now, when the absolute value of the iterated function exceeds a given value R the iteration stops. Then, the pixel becomes white if the absolute value of the real part or the imaginary part of z is smaller than R and else it is black. This results in the self-similar image shown at the left. Here R=5 and the range between -7 and +7 is shown for x and y.

Near the border we have large absolute values of z greater than R. Thus the function f(z) is not evaluated and we simply see four white lines of thickness 2*R parallel to the axis. Inside the circle of radius R we first see eight white lines. They arise because the function f(z) is evaluated once. As we have seen before, f(z)=z*z makes an anamorphic image that has two turns of the original image. This makes eight lines out of the four. Similarly, in the smaller regions near the center, there are anamorphic images of the four lines with an increasing number of lines corresponding to the number of repetitions of f(z). This makes up the self-similar zoo-morphs discovered by Pickover.

Now I want to use any input image instead of these lines. The only problem is that the distance r from the center is not limited when the iteration stops. Thus I cannot make a direct mapping between the x- and y-coordinates and an image of limited size. However, I note that r has to be greater than the limiting radius R and thus I can use the inversion at the circle of radius R to map all points into this circle. From z=x+iy I get then the coordinates x’=x*(R/r/r) and y’=y*(R/r/r) of the point in the original image to look up the color. This defines the anamophosis I have used to get this image:


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