Buddhabrot 1147024 212377834 2008-05-14T16:13:30Z 77.186.237.116 /* Rendering method */ [[Image:buddhabrot-deep.jpg|thumb|Deeply iterated ''Buddhabrot'']] The '''''Buddhabrot''''' is a special rendering of the [[Mandelbrot set]] which, when traditionally oriented, resembles to some extent certain depictions of the [[Gautama Buddha|Buddha]]. When viewed upside-down, it vaguely resembles a human face with large, triangular glasses or goggles over its eyes. ==Discovery== The ''Buddhabrot'' rendering technique was discovered and later described in a 1993 [[Usenet]] post [http://groups.google.com/groups?hl=en&lr=&selm=18778%40autodesk.COM] to sci.fractals by Melinda Green. Previous researchers had come very close to finding the precise Buddhabrot technique. In 1988 Linas Vepstas relayed images of the Buddhabrot 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]]. [[Image:buddhabrot-deep-only.jpg|thumb|''Buddhabrot'' showing only slow escapes]] [[Image:BuddabrothRGB.png|thumb|''Buddhabrot in [[RGB]]'']] ==Rendering method== Mathematically, the Mandelbrot set consists of the [[set]] of points ''c'' in the [[complex number]] plane for which the [[iteration|iteratively]] defined [[sequence]] :<math>z_{n+1} = {z_n}^2 + c</math> with ''z''<sub>0</sub> = 0 does ''not'' tend to [[infinity]]. However, the ''Buddhabrot'' is rendered by creating a 2-[[dimension]]al [[array]] of counters, one for each pixel. Then, a random (or, alternately, 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 colors (or color saturation/brightness) are then chosen based on the values recorded in the array. [[Image:nebulabrot.jpg|thumb|''Nebulabrot'']] ==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 Mandlebrot 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 [http://www.steckles.com/buddha/ 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&nbsp;&mdash; 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''. [[Image:anti-buddabrot.jpg|thumb|''Anti-Buddhabrot'']] ==External links== {{Commons|Buddhabrot}} {{external links}} * [http://www.superliminal.com/fractals/bbrot/bbrot.htm ''Buddhabrot'' discoverer Melinda Green's page] * [http://www.linas.org/art-gallery/mandel/mandel.html ''Buddhabrot'' discoverer Linas Vepstas page] * [http://www.complexification.net/gallery/machines/buddhabrot/ ''Buddhabrot'' page from the ''Gallery of Computation''] * [http://www.mrob.com/pub/muency/buddhabrot.html ''Buddhabrot'' page in the ''Mu-Ency'' Mandelbrot Set Encyclopedia] * [http://cabin.users.geeky.net/Fractal.php Generator for various types of ''Buddhabrot'' fractals including modified functions] * [http://www.fractals.quorq.com/bb/bb.php Realtime Buddhabrot/Nebulabrot Renderer Applet] * [https://sourceforge.net/projects/buddhagen ''Buddhabrot'' generator for Windows XP.] * [http://buddha.sourceforge.net A simple cross-platform command line ''Buddhabrot'' renderer.] * [http://local.wasp.uwa.edu.au/~pbourke/fractals/buddhabrot/ General discussion including parallel (MPI) code for multiprocessor clusters] * [http://usuarios.lycos.es/llopsite/ Personal site on the ''Buddhabrot'' with ''Java'' code + artwork. In Spanish & English.] * [http://www.danvk.org/wp/2007-04-06/nebulabrot/ High resolution renderings of the Nebulabrot] [[Category:Fractals]] [[fr:Buddhabrot]] [[it:Buddhabrot]] [[pt:Buddhabrot]] [[uk:Буддаброт]]