Category Archives: Self-similarity

Coloring the Julia set

The Julia set of a function f(z) in the complex plane has all points z that remain finite upon iterations of the function. In the last posts I have used expanding functions to get fractal images from iteration, as discussed … Continue reading

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Rainbow colors

We can define a continuous number x of iterations needed to reach the critical radius R. Note that if the n-th iteration of f(z) equals R then x=n, and if the (n-1)th iteration equals R then x=n-1. For values in-between … Continue reading

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self-similar fractals with rotational symmetry from function iteration

I was looking at my posts of march 2013 on complex function iterations, see in particular “fractal surprise from complex function iteration” and “self-similar images from iterated mappings of the plane“, and I got some new ideas I want to … Continue reading

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repeated plane mapping, anamorphosis and mirrors

The image of the last post is quite bewildering. To get a calmer and simpler image I imposed mirror symmetries on the original image. A vertical mirror line results from taking the absolute value of the x-coordinate. Similarly, I take … Continue reading

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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. Complex numbers z=x+iy represent the (x,y)-plane and complex functions f(z) define a mapping … Continue reading

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Self-similar images from iterated mappings of the plane

A mapping of the plane defines simply another point (u,v) in the plane as a function of the coordinates (x,y) of a point in the plane. The mapping is defined by the functions for the new coordinates u=f(x,y) and v=g(x,y). … Continue reading

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Complex function iteration: Further results and a puzzle

I used the method of the post “Fractal surprise from complex function iteration” for the function where c is a constant. To get a better image I now use dark blue for all numbers z that grow without limit in … Continue reading

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