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2013-06-09 Variations on Nova
2013-05-12 Two new gigapixel images
2013-04-28 Lattes Julia sets
2012-10-21 3d hyperbolic limit sets
2012-07-01 Fractal sea creatures
2012-06-24 Nicholas A. Cope
2012-06-16 3d printed Julia set
2012-05-06 Pep Ventosa
2012-04-06 Fractal automata
2012-03-30 Periodic Julia patterns
June 9th 2013
Variations on Nova
Dense Julia sets are a surprisingly rich source of fractal patterns. Julia sets are associated to the iteration of conformal maps of the sphere to itself, i.e. transformations of the sphere into itself that preserves the angles on the sphere. When the sphere is modeled by the complex plane plus a point at infinity, such maps are rational maps in one complex variable. By picking a point and iterating the transformation, one gets a sequence of points (an "orbit") on the sphere. The Julia set is the set of initial points whose orbits are chaotic, in the sense that their orbit does not stay close to the orbits of the neighbouring points. Dense Julia sets are associated to transformations such that the orbits of all the points of the sphere are chaotic.
In order to extract a fractal pattern from the Julia set, one has to color each point of the sphere according to a certain property of the orbit. One very simple property is the mean distance of the points of the orbits to a given reference point on the sphere. More generally, one can sum over the orbit the value of some real-valued function on the sphere. As all the orbits are chaotic, and therefore differ sensibly from the neighbouring orbits, the pattern drawn by such coloring methods are very intricate. In fact they are dense fractal patterns, i.e. fractal patterns that display structure everywhere, at every scales.
I recently undertook a systematic exploration of the fractal patterns that can be obtained from a single conformal map on the sphere, known as "Nova". The points were colored in function of their distance to a reference point on the sphere. I also considered integer powers of this map, which amounts to dropping points whose rank in the orbits are not multiples of a given integer. I varied systematically the position of the reference point on the sphere and the power integer, and obtained 105 images, visible in this google+ album.
A few of these pictures can also be seen on this website, as zoomable high resolution images and spherical panoramas (link at the bottom of each image's page). You can recognize the same underlying structure in all images, like for instance four main spirals at the corners of a parallelogram. Yet the details of the fractal pattern are different for each of the 115 images.
It was quite surprising to me that such a wide diversity of patterns can be extracted from a single map, by varying trivial parameters in the way the orbits are colored. Varying a few other equally trivial parameters in the coloring yielded even more new patterns, which might be featured here in the future.
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