{"id":1522,"date":"2020-03-06T11:00:39","date_gmt":"2020-03-06T19:00:39","guid":{"rendered":"https:\/\/blog.ronrecord.com\/?p=1522"},"modified":"2025-10-02T10:32:45","modified_gmt":"2025-10-02T17:32:45","slug":"the-buddhabrot-fractal-mandelbrot-set-and-the-logistic-map","status":"publish","type":"post","link":"https:\/\/blog.ronrecord.com\/index.php\/2020\/03\/06\/the-buddhabrot-fractal-mandelbrot-set-and-the-logistic-map\/","title":{"rendered":"The Buddhabrot Fractal, Mandelbrot Set, and The Logistic Map"},"content":{"rendered":"\n

The Buddhabrot<\/strong> is a fractal<\/a> rendering technique related to the Mandelbrot set<\/a>. Its name reflects its pareidolic<\/a> resemblance to classical depictions of Gautama Buddha<\/a>, seated in a meditation pose with a forehead mark (tikka<\/a><\/em>) and traditional topknot (ushnisha<\/a><\/em>).<\/p>\n\n\n\n[embedyt] https:\/\/www.youtube.com\/watch?v=zxIcydL7wwY[\/embedyt]\n\n\n\n

The Buddhabrot<\/em> rendering technique was discovered by Melinda Green (then known as Dan Green), who later described it in a 1993 Usenet<\/a> post to sci.fractals.[<\/a><\/sup><\/p>\n\n\n\n

Previous researchers had come very close to finding the precise Buddhabrot technique. In 1988, Linas Vepstas relayed similar images[<\/a><\/sup> to Cliff Pickover<\/a> for inclusion in Pickover’s then-forthcoming book Computers, Pattern, Chaos, and Beauty<\/em>. This led directly to the discovery of Pickover stalks<\/a>. 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.<\/p>\n\n\n\n

Green first named this pattern Ganesh, since an Indian co-worker “instantly recognized it as the god ‘Ganesha<\/a>‘ which is the one with the head of an elephant.”[<\/a><\/sup> The name Buddhabrot<\/em> was coined later by Lori Gardi.<\/p>\n\n\n\n

Here is a video by YouTube user Mathologer describing, in a somewhat cheezy fashion, some of the interesting mathematics involved in depicting both the Mandelbrot halo used to generate the Buddhabrot and the complexity within the Mandelbrot Set itself.<\/p>\n\n\n\n[embedyt] https:\/\/www.youtube.com\/watch?v=9gk_8mQuerg[\/embedyt]\n\n\n\n

Some of what YouTube user Mathologer presented in the video above is the beginnings of describing the close relationship between the Mandelbrot Set and the Logistic Map.<\/a><\/strong><\/p>\n\n\n\n

The logistic map<\/strong> is a polynomial<\/a> mapping<\/a> (equivalently, recurrence relation<\/a>) of degree 2<\/a>, often cited as an archetypal example of how complex, chaotic<\/a> behaviour can arise from very simple non-linear<\/a>dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May<\/a>, in part as a discrete-time demographic model analogous to the logistic equation<\/a> first created by Pierre Fran\u00e7ois Verhulst<\/a>.[<\/a><\/sup> Mathematically, the logistic map is written<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

where xn<\/sub><\/em> is a number between zero and one that represents the ratio of existing population to the maximum possible population. The values of interest for the parameter r<\/em> (sometimes also denoted \u03bc<\/em>) are those in the interval [0,4]. This nonlinear difference equation is intended to capture two effects:<\/p>\n\n\n\n