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November 1st 2011
First, an announcement. A few of my works in prints will be on display at Montreux Art Gallery, an art fair in Montreux, Switzerland, from November 9th to November 13th. Look for Galerie Les 3 Soleils, booth 40, level 1, in the gallery section.
Below are two animations based on the "Inverted Julia" fractal. Recall that the Inverted Julia fractal is obtained by a very similary method as the standard Julia fractal. Iterations are performed on the plane of complex numbers (identified with the pixels of the picture) and the pixels are colored depending on the behavior of the corresponding orbits. The main difference is the presence of an inversion transformation, which prevents the orbits from escaping to infinity. The Inverted Julia set also admits a complex number as a seed, which determines the type of patterns in the fractal. Interestingly, for certain values of the seed, the orbits become chaotic, and the algorithm produces space filling fractal patterns. These patterns bear close resemblance to the ones which can be spotted in the Mandelbrot and Julia sets, except for their denseness, which in my opinion add to their beauty. You can admire Inverted Julia patterns in this collection.
In the first animation, the seed of the Julia set is simply moved along a path in the complex plane. It displays the transitions to and from a region of chaotic orbits and space-filling fractal pattern.
The coloring used to reveal the fractal pattern is similar to the famous "orbit trap" method used to color standard Mandelbrot and Julia sets. Each times the orbit lands sufficiently close to the origin (i.e. gets "trapped"), a number related to the distance to the origin is added, and the color is determined by the total sum. What "sufficiently close" means is govered by a parameter, the size of the trap.
In the second animation, the size of the trap is slowly varied from a small value to a large one. At the beginning of the video, the orbits almost never hit the trap, which is very small, so the picture is almost completely black. As the size of the trap is increased, the pattern develops. Finally, near the end, when the trap becomes too big, the orbits always hits it, and the picture turns all white. Varying the trap size yields a surprising diversity of patterns.
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