# Mandelbrot set

Initial image of a Mandelbrot set zoom sequence with a continuously colored environment Mandelbrot animation based on a static number of iterations per pixel remains bounded.[1] That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets. For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules, and is one of the best-known examples of mathematical visualization. History The first picture of the Mandelbrot set, by Robert W. The Mandelbrot set

Benoit Mandelbrot Benoît B. Mandelbrot[note 1][note 2] (20 November 1924 – 14 October 2010) was a Polish-born, French and American mathematician, noted for developing a "theory of roughness" in nature and the field of fractal geometry to help prove it, which included coining the word "fractal". He later discovered the Mandelbrot set of intricate, never-ending fractal shapes, named in his honor.[7] While he was a child, his family fled to France in 1936 to escape the growing Nazi persecution of Jews. After World War II ended in 1945, Mandelbrot studied mathematics, graduating from universities in Paris and the U.S., receiving a masters degree in aeronautics from Caltech. He spent most of his career in both the U.S. and France, having dual French and American citizenship. Because of his access to IBM's computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovering the Mandelbrot set in 1979. Early years A Mandelbrot set

Mathematical visualization The Mandelbrot set, one of the most famous examples of mathematical visualization. Applications Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis. Notable examples include plane curves, space curves, polyhedra, ordinary differential equations, partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such as soap films), conformal maps, fractals, and chaos. Examples Sphere eversion – that a sphere can be turned inside out in 3 dimension if allowed to pass through itself, but without kinks – was a startling and counter-intuitive result, originally proven via abstract means, later demonstrated graphically, first in drawings, later in computer animation. The cover of the journal The Notices of the American Mathematical Society regularly features a mathematical visualization. Software See also Mathematical diagram References External links

Edward Norton Lorenz Edward Norton Lorenz (May 23, 1917 – April 16, 2008)[1][2] was an American mathematician and meteorologist, and a pioneer of chaos theory.[3] He introduced the strange attractor notion and coined the term butterfly effect. Biography Lorenz was born in West Hartford, Connecticut.[4] He studied mathematics at both Dartmouth College in New Hampshire and Harvard University in Cambridge, Massachusetts. From 1942 until 1946, he served as a meteorologist for the United States Army Air Corps. After his return from World War II, he decided to study meteorology.[2] Lorenz earned two degrees in the area from the Massachusetts Institute of Technology where he later was a professor for many years. Two states differing by imperceptible amounts may eventually evolve into two considerably different states ... Awards Work Lorenz built a mathematical model of the way air moves around in the atmosphere. See also Publications Lorenz published several books and articles.

Fractal Figure 1a. The Mandelbrot set illustrates self-similarity. As the image is enlarged, the same pattern re-appears so that it is virtually impossible to determine the scale being examined. Figure 1b. The same fractal magnified six times. Figure 1c. Figure 1d. Fractals are distinguished from regular geometric figures by their fractal dimensional scaling. As mathematical equations, fractals are usually nowhere differentiable.[2][5][8] An infinite fractal curve can be conceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.[7]:48[2]:15 There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Introduction The word "fractal" often has different connotations for laypeople than mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception. History Figure 2.

Misiurewicz point Mathematical notation A parameter is a Misiurewicz point if it satisfies the equations and so : where : denotes the -th iterate of Name Misiurewicz points are named after the Polish-American mathematician Michał Misiurewicz.[1] Note that the term "Misiurewicz point" is used ambiguously: Misiurewicz originally investigated maps in which all critical points were non-recurrent (that is, there is a neighborhood of every critical point that is not visited by the orbit of this critical point), and this meaning is firmly established in the context of dynamics of iterated interval maps.[2] The case that for a quadratic polynomial the unique critical point is strictly preperiodic is only a very special case; in this restricted sense (as described above) this term is used in complex dynamics; a more appropriate term would be Misiurewicz-Thurston points (after William Thurston who investigated postcritically finite rational maps). Quadratic maps which has a single critical point at . If under