two circles

Lately I have played around with inversion at circles trying to find some new kind of fractals. Thus I found a simple mapping that gives interesting designs. They are not fractal, instead overlapping circles appear.

Inversion at a single circle is trivial. If the radius is equal to one and the center lies at the origin, then inversion maps a point z in the complex plane to 1/z. A second inversion gives z again and that’s it. But using inversion at two different circles gives a really interesting dynamics. I use two circles of unit radius, one with the center at the origin. The other center lies at a distance d away on the real x-axis. A point at z goes then first to 1/z and then to d+1/(1/z-d). This combined mapping has two fixed points. For d<2 the fixed points are the intersection points of the circles.

The image of a circle upon inversion is always a circle or exceptionally a straight line (see Thus the combined mapping and its inverse also give a unique circular disk as the image of a circular disk. This is very different to polynomial functions like z*z+d with inverse mappings that give several unconnected regions from a compact region. This splitting of the inverse mapping causes the fractal structure of the Mandelbrot set and similar fractals, for more see “self-similar fractals …“. Thus, this mapping does not give fractals.

To get some images for this combined inversion at circles I simply check the distance r of the point z from the center of the circle after each inversion. If r is lower than some chosen value R then the pixel at the starting point of the iteration gets a color. It will be black if this happened at circle at the origin and white for the other circle. If there is no such event after many iterations then the pixel becomes dark red. This happens for starting points in large regions around the two fixed points.

Note that all pixels at points z of absolute value larger than 1/R immediately become black at the first inversion at the circle at the origin. Thus only pixels closer to the origin are interesting. The results depend strongly on the distance d between the circles and the detection radius R. Often they are rather chaotic superpositions of circles. Here you see a more interesting result for R=0.3 and d=1.807:


You can easily make your own experiments. Simply use the “improved code for fractals …” and modify the “color fractal(Vector z)” function.

Could these images be mistaken as abstract geometric or concrete art ? Naively, one might think that an artist used compasses and ruler to create it. One could try to find complicated geometric rules and would never discover that it simply results from the inversion of points at circles.

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