On Mandelbrot and Fractals The


On Mandelbrot and Fractals

The pictures that you might see around the web of the Mandelbrot set are a kind of haze on a very precise, very interesting mathematical entity. Fractals images are generated around complex numbers--that is a number with a real root and an imaginary root. (recall that the basic definition of an imaginary number i is the square root of negative one--or that number which, when multiplied by itself will yield negative one.)

A number in the complex plane is tested to see if it belongs to the Mandelbrot set. The test is to plug the number into the equation

Z= Z^2+C.

The number you are testing is C. Z starts at zero. You calculate the value using the value of C. Once you arrive at a value, you use the new value to plug back into the equation. Say for example you start with 2i. The answer the first time around to the equation is 2i. The next time you figure your result, you will need to calculate

Z=2i^2+ 2i, or -4+2i

For a point to be part of the Mandelbrot set (the dark area of any Mandelbrot diagram) the magnitude of the value of Z (the absolute value) must never exceed 2. Thus, with our example, 2i is not a point on the Mandelbrot set, nor is it likely even to be in an interesting area around the set.

Now, some points will never exceed 2; for example, 0i will never exceed 2, because the result will always be 0. If results do exceed 2, they will rapidly increase to infinity. You assign colors to points based on how rapidly the value of Z increases in that region. Thus you get the highly colored, multi-textured diagrams of the Mandelbrot set. Of course, the Mandelbrot set is only the most famous body of fractals. There are all sorts of ways to generate fractals, including such things as julia sets, cantor dusts, sierpinski gaskets and triangles, koch curves, and newtonian basins of atrraction. Each yields a unique and beautiful pattern. Anyway, I hope that this brief introduction has sufficiently impressed you that you feel free to go forth and explore this amazing mathematical world.

Click here for an image of a fractal from a private page. Click here to see the page and get perhaps a clearer explanation for what is happening.

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This page contains a single entry by Steven Riddle published on February 9, 2003 4:46 PM.

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