LYCOS RETRIEVER 
Mandelbrot Set: Points
built 295 days ago
The Mandelbrot set is actually the region that you see colored in black when the applet first starts. (You can change the color using the "Mandelbrot Color" menu.) More exactly, black points are possibly in the set, but for some black points, more computation would show that they are not really in the set. Colored points are definitely outside the Mandelbrot set. The color depends on the number of "iterations" that are computed before it is determined that the point is not in the set. Points that are closer to the set require a larger number of iterations. The "MaxIterations" menu determines how many iterations the program will compute for a given point before giving up and coloring that point black.
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The basic algorithm to generate the Mandelbrot set is as follows. For each pixel c, start with Z = 0. Iterate the above equation up to N times, exiting if |Z| gets large. If you finish the loop, the point is probably inside the Mandelbrot set. If you exit, the point is outside and can be colored according to how many iterations were completed. You can exit if |Z| > 2, since if Z gets this big it will go to infinity.
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Mandelbrot was looking for some sort of clue as to which c numbers made disconnected sets, and which made connected sets. It turns out that the test is easy. You just start with a z of 0+0i, the origin of the complex plane. This is called a "critical" point for this equation. If this point is class (1), the Julia set is of the dust type. If this point is class (2), the Julia set is of the solid type.
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It's quite obvious that the outer bands around the Mandelbrot set form complete loops around the Mandelbrot set. The band representing two iterations for instance, travels smoothly around the outer edges and then connects back up to itself. There are no other points that have an iteration count of two except for on this band, and all the points on this band are connected by other points with an iteration count of two. This is less obvious but equally true for all other bands. If you zoom in to the band that represents ten iterations, you can drive all the way around the Mandelbrot set, following that band, and return to where you started. You could try it for the band representing one hundred, or one thousand iterations, but it would take a very long time.
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The Mandelbrot set is self-similar under magnification in the neighborhoods of the Misiurewicz points. It is ... conjectured to be self-similar around generalized Feigenbaum points (e.g. -1.401155 or -.1528 + 1.0397i), in the sense of converging to a limit set.[9][10]
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What is the Mandelbrot set? A mathematician might say it was the locus of points, C, for which the series Zn+1 = Zn * Zn + C, Z0 = (0,0) is bounded by a circle of radius two, centered on the origin.
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