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Mandelbrot Set: Fractals
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The Mandelbrot Set is a well-known fractal, with an infinitely detailed structure that is highlighted by computer generated colors. It can be magnified by enormous amounts and still keep its detail. (If you have tried zooming in with your program and the detail smooths out after a while click here.) Formula Details
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At three-and-a-half hours in duration, The Amazing Mandelbrot Set is a truly unique creation. It's the only feature-length exploratory video catalogue of fractal geometry ever produced--and it's available worldwide for the low price of only $22 USD, plus shipping.
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"The Mandelbrot set can be divided into an infinite set of black figures: the largest figure in the center is a cardioid. There is a (countable) infinity of near-circles (the only one to be actually an exact circle being the largest, immediately on the left of the cardioid) which are in direct (tangential) contact with the cardioid, but they vary in size, tending asymptotically to zero diameter. Then each of these circles has in turn its own (countable) infinite set of smaller circles which branch out from it, and this set of surrounding circles ... tends asymptotically in size to zero. The branching out process can be repeated indefinitely, producing a fractal. Note that these branching processes do not exhaust the Mandelbrot set: further upwards in the tendrils, some new cardioids appear, not glued to lower level "circles". The largest of these, and the most easily visible from a view of the entire set, is along the "spike" which follows the negative real axis out, roughly from real values of -1.78 to -1.75."
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The Mandelbrot set, named after Benoit Mandelbrot, is a fractal. Fractals are objects that display self-similarity at various scales. Magnifying a fractal reveals small-scale details similar to the large-scale characteristics. Although the Mandelbrot set is self-similar at magnified scales, the small scale details are not identical to the whole. In fact, the Mandelbrot set is infinitely complex. Yet the process of generating it is based on an extremely simple equation involving complex numbers.
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Mandelbrot discovered that the fourth dimension of fractal forms includes an infinite set of fractional dimensions which lie between the zero and first dimension, the first and second dimension and the second and third dimension. He proved that the fourth dimension includes the fractional dimensions which lie between the first three. He calls the in between or interval dimensions the "fractal dimensions." Mandelbrot coined the word fractal based on the Latin adjective "fractus." He choose this word because the corresponding Latin verb "frangere" means "to break," "to create irregular fragments." He has shown mathematically and graphically how nature uses the fractal dimensions and what he calls "self constrained chance" to create the complex and irregular forms of the real world.
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The main purpose of the Mandelbrot set is to index Julia sets corresponding to various values of the parameter c. When c belongs to the Mandelbrot set, Jc is connected. For c outside M, Jc is totally disconnected and known as the fractal dust. Both M and Jc are visualized with a simple algorithm that assigns a color value to a pixel depending on how fast it was found out whether iterations for that pixel escape to infinity. Since for c inside M, the iterations remain bounded, pixels corresponding to the Mandelbrot set consume the greatest amount of the computational time.
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Fractals