The Buddhabrot is a special rendering of the Mandelbrot set which, when traditionally oriented, resembles to some extent certain depictions of the Buddha. When viewed upside-down, it vaguely resembles a human face with large, triangular glasses or goggles over its eyes.
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Discovery
The Buddhabrot rendering technique was discovered and later described in a 1993 Usenet post [1] to sci.fractals by Melinda Green
Previous researchers had come very close to finding the precise Buddhabrot technique. In 1988 Linas Vepstas relayed similar images to Cliff Pickover for inclusion in Pickover's forthcoming book Computers, Pattern, Chaos, and Beauty. This led directly to the discovery of Pickover stalks. These researchers did not filter out non-escaping trajectories required to produce the ghostly forms typically reminiscent of Hindu art. Green first named it Ganesh, since an Indian co-worker "instantly recognized it as the god 'Ganesha' which is the one with the head of an elephant." The name Buddhabrot was coined later by Lori Gardi.
Rendering method
Mathematically, the Mandelbrot set consists of the set of points c in the complex number plane for which the iteratively defined sequence
where z0 = 0 does not tend to infinity.
The Buddhabrot is rendered by first creating a 2-dimensional array of counters, each counter corresponding to a final pixel of the image and initialized to zero. Then, a random sampling of c points are iterated through the Mandelbrot function. For points which do escape within a chosen number of iterations, and therefore are not in the Mandelbrot set, their values are sent through the Mandelbrot function again and this time its path is plotted into the array. After a large number of c values have been iterated, grayscale shades are then chosen based on the values recorded in the array. The result is a density plot highlighting regions where z values spend the most time on their way to infinity.
Nuances
Because rendering Buddhabrot involves potentially iterating twice over each sample (once to test if it escapes, and again to plot its path if it does), it is more computationally intensive than standard Mandelbrot rendering techniques. To add to this, rendering highly zoomed areas requires even more computation, as the path of an escaping point may enter the portion being rendered from outside. Without resorting to more complex probabilistic techniques, rendering zoomed portions of Buddhabrot consists of merely cropping a large full sized render.
The number of iterations chosen has a large effect on the image — higher values give sparser more detailed appearance, as a few of the points pass through a large number of pixels before they escape, resulting in their paths being more prominent. If a lower number of iterations was used, these points would not escape in time and would be regarded as not escaping at all.
It is also possible to create a composite from three images with different numbers of iterations and different colours; for example, combining a red image with 2,000 iterations, a green image with 200, and a blue image with 20, a technique similar to how astronomers produce false-color images. Some have labelled this the Nebulabrot as it results in a very Nebula-like image.
Another technique which it is natural to consider is to plot the paths for points c which are in the Mandelbrot set; a sort of Anti-Buddhabrot.
Renderings
The Buddahbrot can be seen as an overlay convergence-image of maps of the complex plane onto itself under different powers of the Mandelbrot iteration z2+c. Therefore, a natural way of coloring is to use a color map of the complex plane, then do successive iterations and pile color information in the overlay image. This tracks to some extent which points from the original complex plane result in which structures in the Buddhabrot fractal. In order to obtain a good image, one can use minute random perturbations in the starting point positions so that several passes generate a stable image. The Complex Map Buddhabrot image shows the result of applying this technique on all points. Besides the max number of iterations used for rendering the image, the initial z0 value used also strongly influences the image. Whereas the classical Buddhabrot set originates from the initial value z0 = 0+0i, the image for z0 = 0.2+0.5i resembles a heart-shaped form.
External links
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Wikimedia Commons has media related to: Buddhabrot |
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This article's external links may not follow Wikipedia's content policies or guidelines. Please improve this article by removing excessive or inappropriate external links. |
- Buddhabrot discoverer Melinda Green's page
- Buddhabrot discoverer Linas Vepstas page
- Buddhabrot page from the Gallery of Computation
- Buddhabrot page in the Mu-Ency Mandelbrot Set Encyclopedia
- Generator for various types of Buddhabrot fractals including modified functions
- Realtime Buddhabrot/Nebulabrot Renderer Applet
- Buddhabrot generator for Windows XP.
- A simple cross-platform command line Buddhabrot renderer.
- General discussion including parallel (MPI) code for multiprocessor clusters
- Personal site on the Buddhabrot with Java code + artwork. In Spanish & English.
- High resolution renderings of the Nebulabrot
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