Recently I got the inexpensive Dover reprint of Clifford A. Pickover’s book “Computers, Patterns, Chaos and Beauty”. Part of it extends topics presented in “The Armchair Universe” by Dewdney. And there are other interesting ideas in Pickover’s book. Get it, it is really very interesting.
There you can find a good discussion of iterations of complex function like f(z)=z*z+c; where c is a constant. From the absolute values of the iterated function one gets the well-known Julia and Mandelbrot sets. Recently I looked at the phase of the complex numbers instead.
I am trying to explain. But beware, I am not a good writer and not really good in complex analysis.There will be a lot of equations. If you don’t like that, feel free to skip to the images at the end of the post.
The two-dimensional plane can represent complex numbers and functions. A point with coordinates x and y corresponds to the complex number z=x+i*y, where i is the imaginary unit with i*i=-1. Just thinking of polar coordinates we note that r=sqrt(x*x+y*y) is the absolute value of z. Its phase φ is defined by z= r*cos(φ) + i*r*sin(φ). Thus φ=tan(y/x), where care has to be taken to use the correct branch of the tangent function.
Now, the phase φ is a cyclical variable as φ+2*π is essentially equal to φ. Similarly, the hue is cyclical, going from red to green to blue and to red again. Thus we can represent φ=0 by red, φ=2*π/3 by green and φ=4*π/3 by blue. Note that φ=2*π is red again. For intermediate values corresponding mixtures are taken.
The square of a complex number is simply z*z=x*x-y*y+i*2*x*y. From the multiplication of trigonometric functions we get that in polar coordinates z*z=r*r*cos(2*φ)+i*r*r*sin(2*φ). Thus the absolute value squares and the angle φ doubles. Now, for large z we get a trivial change in the phase by the function f=z*z+c as the phase φ is simply augmented by itself. These trivial changes are summed up at each iteration step and finally subtracted from the phase of the iterated function value. This eliminates a potentially infinite number of color changes for large numbers z and is inspired by the renormalization of singularities in particle physics. I will publish the detailed program code in the next post. This explains more clearly what I am doing and you will be able to experiment yourself.
For different values of the constant c we get rather different images. But they all have a fractal structure. There are many points where all color hues meet. These singular points themselves have no defined color. I suppose that they arise, when the function f becomes zero at some iteration step. These zeros are distributed in a self-similar fractal manner.
Large values of the constant c give a hierarchical pattern of spirals:
Decreasing values for c lets the spirals unroll until they are straigtened out:
Finally, for very small c we get discs with a fractal branching pattern. It somehow resembles images of Hexacoralla, such as the Stephanophyllia Elegans, as shown in Ernst Haeckel’s book “Art Forms in Nature”: