Mandelbrot SetDate: 10/22/96 at 14:33:17 From: Justin Rhees Subject: What is the Mandelbrot set? I would like to know the definition of the Mandelbrot set and what it does. Thank you. Date: 10/23/96 at 17:24:17 From: Doctor Anthony Subject: Re: What is the Mandelbrot set? You will need to know a little about complex numbers to understand the Mandelbrot set. A complex number z is given by z = x + iy where i is sqrt(-1) and x and y are real numbers. The Argand diagram has the usual x and y axes, with REAL numbers plotted along the x axis and the y numbers called IMAGINARY plotted along the y axis. So if z = 3 + 4i, then z would be plotted at the point (3,4), and would (by Pythagoras) be 5 units from the origin. This number 5 is called the modulus of the complex number. The Mandelbrot set is a portion of the Argand diagram which satisfies a particular condition. To test whether a particular complex number c is in the set, we carry out the following iteration, starting at z = 0: z1 = (z0)^2 + c z2 = (z1)^2 + c z3 = (z2)^2 + c . . . . . and on and on and on! This iteration is continued for as long as is necessary to see if z is heading off to infinity. If z begins to move further and further from the origin, then the point c does not belong to the set. If x or y becomes greater than 2 or less than -2, it is surely heading off to infinity. But if the program repeats the calculation many times (thousands if necessary) and neither the real or imaginary or real part becomes greater than 2, then the point c is part of the set. The program is repeated for every point c of the complex plane (in practice thousands of points on a grid), and the results are displayed. Points in the set can be colored black, other points white. For a more vivid picture, the white points can be replaced by colored gradations. If the iteration exceeds 2 after 10 repetitions, for example, the program might plot a red dot; for 20 repetitions an orange dot; for 40 repetitions a yellow dot and so on. The colors reveal the contours of the terrain just outside the set proper. The resulting shape is remarkable for its intricate and curious geometry. It has been described as the most complex mathematical shape ever invented - yet you can get a computer to draw it with about ten lines of program code. The most startling feature of the Mandelbrot set is the way it retains its highly complicated structure if you zoom in on it at ever higher levels of magnification. It is infinitely scalable, so that even after enlargements of many millions, it shows the same structure of whirlpools, scrolls, seahorses, lumps, sprouts, cacti, coils, blobs and zigzags. And every so often, buried deep within the structure, perhaps a millionth of the size, you can find an exact replica of the original shape, complete in every detail together with its own replicas at an even deeper level. Standard geometry takes an equation and asks for the set of points that satisfy it. Thus we obtain simple equations for circles, ellipses, parabolas and straight lines. But if we iterate an equation instead of solving it, the equation becomes a preocess instead of a description, dynamic instead of static. When a number goes into the equation, a new number comes out; the new number then itself goes in and so on, points hopping from place to place. A point is plotted not when it satisfies the equation, but when it produces a certain kind of behaviour. One behaviour might be a steady state. Another might be a convergence to a periodic repetition of states. Another might be an out-of-control race to infinity. With computers, trial and error geometry of this sort became possible. The Mandelbrot set is the boundary between two types of radically different patterns and is a model for chaotic behaviour. For a good online introduction to fractals, please see: http://ieng9.ucsd.edu/~esuddert/fractal/index.html -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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