LYCOS RETRIEVER 
Mandelbrot Set
built 254 days ago
The Mandelbrot Set is a fractal, an infinitely detailed mathematical form generated by a formula. In this case the formula is as follows: Given the x and y coordinates of a pixel, record them as j and k. Repeatedly do this: replace x with x2-y2+j and y with 2xy+k. See if the numbers have grown really huge. Count the number of times this has been repeated. If the numbers reach the point that x2+y2 is more than 4 stop the iterating (repeating) and color the point on the screen based on the number of iterations. Since sometimes the numbers never get large enough, if the number of iterations passes a maximum, stop and color the point black.
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The Mandelbrot Set is a fractal named after its discoverer, Benoit Mandelbrot. Mandelbrot coined the term "fractal" in 1975 from the Latin fractus or "to break". Fractals are things that are self-similar at various scales. Magnification of a fractal will reveal small details similar to larger characteristics. In the Mandelbrot Set, these small details do not replicate the larger whole exactly, and ... it is said to have only quasi self-similarity. The Mandelbrot Set is a fractal created by a very simple mathematical formula:
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When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies the boundary as a fractal.
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The Mandelbrot Set emerges from the behavior of a famously simple mathematical function. The Set itself is like a black hole in the abstract space it inhabits. Most of that space is a vast, featureless void. But the points near the boundary of the Set are torn between the temptation to join the Set and the lure of infinity. When their behavior is coded in colors, the result is a beautiful filigree of infinite depth and complexity.
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The Mandelbrot Set consists of all those complex numbers C for which a certain sequence of does not diverge (in other words, the values in the sequence do not get infinitely large). The sequence is a simple one: zt+1 = z tk + C, where k=2 and z0=0+0i. If, after an infinite number of applications of this formula, the magnitude |z | stays small, then the number C is a member of the Mandelbrot Set. Otherwise, C is not a member.
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The Mandelbrot Set is defined by the iteration of a formula applied to all of the points on the complex plane. For any point (x, y) in a given image, that point is represented by a complex number c=a+bi. The formula z←z2+c is iterated (with an initial value for z of 0) to see what happens to z. If z grows larger and larger without bound (converging on infinity), then it is not part of the Mandelbrot Set, and is colored according to how quickly it converges towards infinity. If z stays bounded over some given iteration limit, then it is considered to be part of the Mandelbrot Set, and is colored with a chosen "set" color (typically black). These images are composed almost entirely of points that are not part of the set itself, but that converge towards infinity at different speeds, yielding very colorful panoramas.
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Mandelbrot Set