Here is a zoom on a Ducks Julia about the origin, using Damien Jones' whoosh flash applet. except if you're zooming about zero! You can zoom as deep as you want and keep the same level of detail with the same number of iteration. The same is true about the Ducks Julia fractal. In the standard Mandelbrot fractals, the more you zoom, the more iterations you have to perform to keep the same level of details. For aesthetic reason, I turned the pattern by 90° so that the real axis is vertical.įive examples of Ducks Julia-type fractals, chosen among those that fills the plane densely with fractal patterns. Here are five choices of c, bundled up in an animated gif. Now if we choose c to lie in the middle region where the intricate structures lie, it turns out that we get infinitely extended fractal patterns that fill the plane densely. Similarly, we can draw Julia Ducks fractals by replacing p by a fixed number c in the algorithm above. ![]() Remarkably, the Julia set based on c displays structures similar to the ones that can be seen in the Mandelbrot set around c. For each point c in the complex plane, there is a Julia set, that we can draw by iterating the same function, but replacing p by the number c, that is fixed for all pixels. Indeed, to draw the Mandelbrot set, we have to iterated z = z^2 + p. ![]() Note that this fractal involves iterating a function that depends on the original pixel coordinates p. ![]() It might not be obvious in the picture above, but there are some very intricate structures between the two curved arcs. The top right corner of the structure lies at 1, while the bottom left lies at -1 - pi*i. If you implement this algorithm, you will get something like this: This type of algorithm, where a function is iterated, is called a dynamical system in mathematics and is a well known source of fractals. Color the pixel corresponding to p according to the mean of the magnitude of z, summed over all iterations. Here the branch cut of the logarithm has been chosen to lie on the negative real axis. Now for each complex number p in the complex plane, iterate the following function for a fixed number of iterations (typically 50 to 100): Geometrically, this transformation replace the lower half complex plane by the mirror image of the upper half complex plane. Let z be a complex number and define Iabs(z) to be equal to z if the imaginary part of z is positive and its complex conjugate if it is negative. Compared to some other Ducky-like algorithms has the nice property that the patterns are never stretched, and that no branch cut ever appears. Because I like Ducky and lack imagination for names, I called it Ducks. In an attempt to simply the algorithms as much as possible in order to understand how they are working, I ended up with the following algorithm that reproduces most of the patterns. Then Ed Algra wrote a few formulas called "Ducky" for Ultra Fractal that reproduced the same type of patterns. I don't know who discovered them, but for some time it was popular among users of Fractal Explorer, with a formula called "Thalis". Some beautiful images using the same type of patterns can be found in misterxz gallery at deviantart.ĭense fractal patterns are exactly what I am after, so I was naturally very interested in these new techniques. ![]() See this picture or this one for typical examples. Here I just want speak a bit about the algorithm.Ī few years ago, a new type of fractal imagery appeared (or at least I hadn't encountered it before), involving fractal patterns apparently filling densely the plane. It is still related to pattern piling, but I'll explain this in another post. Recently I've been using a different type of algorithm. Algorithmic worlds - Blog Algorithmic worldsĪ blog about algorithmic art and fractal aesthetic.
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