Mandelbrot Set

by Tuğrul Yazar | October 10, 2012 13:05

Mandelbrot set is the continuation of a previous study on Julia sets (studied here). According to this website[1]:

The Mandelbrot set, named after Benoit Mandelbrot, is a fractal. Fractals are objects that display self-similarity at various scales. Magnifying a fractal reveals small-scale details similar to the large-scale characteristics. Although the Mandelbrot set is self-similar at magnified scales, the small scale details are not identical to the whole. In fact, the Mandelbrot set is infinitely complex. Yet the process of generating it is based on an extremely simple equation involving complex numbers.

Above is the complete set containing all Julia sets (all C constants that do not escape to infinity). Again I’m stuck with this brute-force approach and cannot see the fractals at the boundary clearly. I think in native Grasshopper, it would not be possible. Therefore, I am only displaying points that meet certain mathematical criteria. However, the real solution is called the inverse iteration method. In this method, instead of putting many points and checking if they are inside the set or not; this approach directly calculates the boundary (limit) of the expression. When I zoom in on the boundary of the set, I barely understand that it really repeats itself and travels to infinity. There are interesting animations on the internet about that. Here is my Grasshopper definition:

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To zoom in; modify the A and B inputs of the first Rectangle component (Rec 2Pt), but be careful about the nr. of points parameter or you may crash your computer. Surprisingly, the definition is so simple and only using core Grasshopper components, it still works in 2021. But I had to wait for a couple of minutes to finish the computation.

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Source URL: https://www.designcoding.net/mandelbrot-set/

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