Мандельброт фракталынан қашқан нүктелердің траекторияларының ықтималдық таралуы

Probability distribution over the trajectories of points that escape the Mandelbrot fractal

The Buddhabrot is the probability distribution over the trajectories of points that escape the Mandelbrot fractal. Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehead mark (tikka), a traditional oval crown (ushnisha), and ringlet of hair.

The Buddhabrot rendering technique was discovered by Melinda Green, who later described it in a 1993 Usenet post to sci. fractals. to Cliff Pickover for inclusion in Pickover's then forthcoming book Computers, Pattern, Chaos, and Beauty. This led directly to the discovery of Pickover stalks. Noel Griffin also implemented this idea in the 1993 "Mandelcloud" option in the Fractint renderer. However, these researchers did not filter out non escaping trajectories required to produce the ghostly forms reminiscent of Hindu art. The inverse, "Anti Buddhabrot" filter produces images similar to no filtering. Green first named this pattern 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 Buddhabrot images is typically more computationally intensive than standard Mandelbrot rendering techniques. This is partly due to requiring more random points to be iterated than pixels in the image in order to build up a sharp image. Rendering highly zoomed areas requires even more computation than for standard Mandelbrot images in which a given pixel can be computed directly regardless of zoom level. Conversely, a pixel in a zoomed region of a Buddhabrot image can be affected by initial points from regions far outside the one being rendered. Without resorting to more complex probabilistic techniques, rendering zoomed portions of Buddhabrot consists of merely cropping a large full sized rendering. The maximum number of iterations chosen affects 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 maximum was used, these points would not escape in time and would be regarded as not escaping at all. The number of samples chosen also affects the image as not only do higher sample counts reduce the noise of the image, they can reduce the visibility of slowly moving points and small attractors, which can show up as visible streaks in a rendering of lower sample count. Some of these streaks are visible in the 1,000,000 iteration image below. Green later realized that this provided a natural way to create color Buddhabrot images by taking three such grayscale images, differing only by the maximum number of iterations used, and combining them into a single color image using the same method used by astronomers to create false color images of nebula and other celestial objects. For example, one could assign a 2,000 max iteration image to the red channel, a 200 max iteration image to the green channel, and a 20 max iteration image to the blue channel of an image in an RGB color space. Some have labelled Buddhabrot images using this technique Nebulabrots.