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. 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) was an American mathematician and meteorologist, and a pioneer of chaos theory. He introduced the strange attractor notion and coined the term butterfly effect. Biography Lorenz was born in West Hartford, Connecticut. 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. 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. 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.:48:15 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. 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. 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
Fractals Fractals From Wikibooks, open books for an open world The latest reviewed version was checked on 19 March 2014. There are template/file changes awaiting review. Jump to: navigation, search “What I cannot create, I do not understand.” — Richard P. Introduction Here you can find algorithms and examples of source code for drawing fractals. Multiplatform, open source and free tools are suggested. Make a good description of programa/algorithms. Try to disjoin computing parameters from creating images ( in other words : "separate the calculation phase from the colouring phase" Claude Heiland-Allen) It can slow the program but makes it easier to understand the algorithm ). If it is possible make one-file programs. Contents Retrieved from " Subjects: Hidden categories: Navigation menu Personal tools Namespaces Variants Views Actions Navigation Community Tools Languages Sister projects Print/export This page was last modified on 19 March 2014, at 15:02.
Fractals and Complexity Fractals and Complexity How would you characterize the images on this page? Describing them using traditional features such as "size" and overall "shape" wouldn't really say much, although it could be very informative for some other forms, such as when characterizing "a 10 cm, round orange" or "a 300 cm x 100 cm oblong watermelon", for instance. But using such descriptors for the images here would oversimplify the detail in their patterns. Describing these patterns using the terms of fractal analysis with FracLac, however, can convey some of the complexity inherent in their design. These images show diffusion limited aggregation, which is a type of fractal growth that can be analyzed with FracLac. What is Fractal Analysis? Fractal analysis is a contemporary method of applying nontraditional mathematics to patterns that defy understanding with traditional Euclidean concepts. In essence, it measures complexity using the fractal dimension. Fractal Patterns Imagine a real fractal.
Bit By Bit, 'The Information' Reveals Everything The InformationBy James GleickHardcover, 544 pagesPantheonList Price: $29.95 We can see now that information is what our world runs on: the blood and the fuel, the vital principle. It pervades the sciences from top to bottom, transforming every branch of knowledge. Information theory began as a bridge from mathematics to electrical engineering and from there to computing. What English speakers call "computer science" Europeans have long since known as informatique, informatica, and Informatik. "The information circle becomes the unit of life," says Werner Loewenstein after thirty years spent studying intercellular communication. Economics is recognizing itself as an information science, now that money itself is completing a developmental arc from matter to bits, stored in computer memory and magnetic strips, world finance coursing through the global nervous system. And atoms? And then, all at once, they did. How much does it compute?
Dimensiune Hausdorff În cadrul topologiei, dimensiunea Hausdorff este un număr real pozitiv, asociat unui spațiu metric și extinde noțiunea de dimensiune a unui spațiu vectorial real. A fost introdusă în 1918 de către Felix Hausdorff și dezvoltată ulterior de către Abram Samoilovici Bezicovici, de unde și denumirea de dimensiune Hausdorff-Bezicovici. Definiție[modificare | modificare sursă] Triunghiul lui Sierpinski, un spaţiu având dimensiunea fractală ln 3/ln 2, ori log23, care este circa 1,58. Dimensiunea Hausdorff ne oferă un mijloc uzual de calculare a dimensiunii unui spațiu metric. Exemplu[modificare | modificare sursă] Determinarea dimensiunii Hausdorff pentru intervalul Pentru Pentru , fie numărul natural astfel ales încât Cu acoperirea specială pentru Urmează Deoarece , avem: Cum însă intervalul acoperă, suma tuturor diametrelor va fi cel puțin 1: Rezultă: Deci: Pentru : Considerând cele două cazuri anterioare, obținem: Așadar: Cazuri concrete[modificare | modificare sursă] Bibliografie[modificare | modificare sursă]