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 write about now.
The well-known fractal Julia and Mandelbrot sets contain the points z of the complex plane for which a suitable iterated function f(z) remains finite. Thus I thought that only contracting functions would give fractal images. But then I noted that the zoomorphic images of Pickover’s book “Computers, Patterns, Chaos and Beauty” lie actually in the part of the complex plane where the iterated function diverges to infinity. This made me think of functions that have diverging iterations everywhere except at their zeros.
An example is
This a simple polynomial, similar to the ones used for making Mandelbrot and Julia sets. It has a zero at z=0 as f(0)=0. But for c>>1 the function is expanding, f(z) >> z for small z, and this zero is unstable. In fact, for most z the iterations of this function will diverge towards infinity. The only exceptions are those z for which some function iteration will exactly hit the origin, f(f(f(…(z)…)))=0. An image of these singular points is not too interesting. To get more, I calculate for each point z the number N of iterations needed to have a larger absolute function value than some critical radius R, |f(f(f(…(z)…)))|>R. Then, depending whether N is even or odd, the pixel that corresponds to z will have one of two colors.
Here is a result for n=6, c=-1.3 and R=sqrt(1.8):
How does this arise ? An important idea is that around any point z we see a transformed copy of the region around its function value f(z). Then, look at zeros f(f(…(z)…))=0 of the iterated function. They are mapped on the center z=0 of the image. If the function f is expanding enough we thus get a reduced and distorted copy of the entire image around each of these zeros.
Where are these zeros ? Let us first look at zeros of f(z) without iteration. We rewrite it as
and see that its zeros are the trivial z=0 and the remaining (n-1) zeros are
This explains why we get an image with five-fold rotational symmetry for n=6. The function f expands the region around the zeros by a factor of (-n*c) and is strongly expanding. Thus these zeros are centers of the largest copies of the entire image.
Essentially, the image is a concentric repetition of a ring-like motif in geometric progression. This arises because approximately f(z)=c*z for small z. Then also f(f(z))=c*c*z and so on. This causes the repetition in radial direction towards z=0.
In the next posts, I will give you the code for generating such pictures and doing your own experiments.