Buddhabrot
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The Buddhabrot is a special rendering of the Mandelbrot set, which resembles to some extent certain depictions of the Buddha (when rotated 90 degrees).
Discovery
The Buddhabrot rendering method was independently discovered by several people. Linas Vepstas relayed images of the Buddhabrot to Cliff Pickover in 1988 for inclusion in Pickover's forthcoming book. This lead directly to the discovery of Pickover stalks. The Buddhabrot was independently discovered and later described in a Usenet post [1] to sci.fractals by Daniel (later known as Melinda) Green in 1993, who wrote:
- If I were a religious person I would certainly take this as some sort of sign.
However, 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
with z0 = 0 does not tend to infinity.
However, the Buddhabrot is rendered by creating a 2-dimensional array of counters, one for each pixel. Then, a random (or alternatively, an evenly spaced) sampling of points c is iterated through the Mandelbrot function, and, for points which do escape within a chosen number of iterations, and are thus not in the Mandelbrot set, the counters for each pixel that the z value landed on are incremented (once per hit). After a large number of values c have been iterated, image colours are then chosen based on the values recorded in the array.
Nuances
The number of iterations chosen has a large effect on the image - higher values give more details and a more "religious" feel, 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 possible to generate an image that shows only the paths of points c which took a long time to escape, with quick escapers not being rendered at all. This removes the cloudy effect but gives a very detailed image.
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. 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.