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The Mandelbrot set is the set of complex pointscthat do not escape to infinity when repeatedly applying the mappingz->z^{2}+c, which admittedly doesn't sound very interesting. (We might as well definef(z) =z^{2}+cto avoid having to type "the mapping" again and again; and this is as good a time as any to mention that we always start the iteration withf(0).)Suppose we start with

c= 0. The first iteration offgivesf(0) = 0^{2}+ 0 = 0; the next iteration givesf(f(0)) =f(0) = 0; next isf(f(f(0))) = 0, and so forth: whencis zero, all iterates offare equal to zero.Next suppose we choose

c= 1. This gives us

f(0) = 0^{2}+ 1 = 1,f(1) = 1^{2}+ 1 = 2,f(2) = 2^{2}+ 1 = 5,f(5) = 5^{2}+ 1 = 26,f(26) = 26^{2}+ 1 = 677,and so forth: the values ultimately escape to infinity.

When we choose

c= -1, we get

f(0) = 0^{2}- 1 = -1,f(-1) = (-1)^{2}- 1 = 0,f(0) = 0^{2}- 1 = -1,so the values alternate between -1 and 0, and the iteration does not escape.

For any

c, the first few iterations offare {0,c,c+c^{2},c+ (c+c^{2})^{2},c+ (c+ (c+c^{2})^{2})^{2}, ...}, and it turns out that the boundary of the set of complex values ofcthat do not escape is very complicated. It can be shown that if the absolute value of an iterate offbecomes larger than 2, the iteration escapes to infinity; the following is a graph of the level curves (|f^{n}(z)| = 2) of the successive iterates off.

Show[ MapIndexed[ Graphics[ ContourPlot[Abs[#], {a, -2.0, 0.5}, {b, -1.2, 1.2}, ContourShading->False, Contours->{2.0}, ContourStyle :> Hue[#2[[1]]/11.0], DisplayFunction->Identity, PlotPoints->123]]&, NestList[#^2 + a + b I &, a + b I, 10]], DisplayFunction->$DisplayFunction, AspectRatio->Automatic];

The simplest way to draw filled Mandelbrot sets seems to be to compute the iterates off, using a more or less fine grid of complex values ofc, coloring each point according to the number of iterations it took for the values to escape to infinity (exceed 2, that is). (At least that's the way I did it here.)Designed and rendered using

Mathematica3.0 for the Apple Macintosh.

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