Fractals, Complex Numbers, and ChaosDate: 01/20/97 at 19:18:18 From: Lua Choon Ngee Subject: Fractals I am reading up on fractals. It seems that they use complex numbers to plot a mathematical formula (such as the Julia set of fractals). Actually, I have not studied complex numbers (such as -2^.5). Therefore, I would like to ask about complex numbers and formulas for fractals. I have the formula for the Mandelbrot Set and the Julia Set but other that that, I do not have any others. Also, is chaos theory related to fractals in any way? Date: 01/21/97 at 23:18:30 From: Doctor Toby Subject: Re: Fractals A lot of questions! Fractals don't necessarily have anything to do with complex numbers. But complex numbers are a convenient way to get points on a plane. A complex number takes the form a + ib, where i^2 = -1. So you can add them: (a + ib) + (c + id) = (a + c) + i(b + d). And you can multiply them: (a + ib) (c + id) = ac + i(ad + bc) + i^2 bd; because i^2 = -1, this is equal to (ac - bd) + i(ad + bc). Just as you can plot real numbers on a line, you can plot complex numbers on a plane; a + ib has coordinates (a,b). There's more to complex numbers involving taking roots and logarithms and stuff like that, but you don't need that for things like Julia sets. Here's a familiar formula for a new fractal you may not know: Compare the formulas for the Mandelbrot set and the Julia sets. The Mandelbrot set says start with z = 0 and repeat the process of replacing z with z^2 + c. If the iteration is bounded, c is in; otherwise not. The Julia set associated with the complex number c says start with z = k and repeat the process of replacing z with z^2 + c. If this iteration is bounded, then k is in the set. You can combine these sets into one giant 4 dimensional fractal. Start with z = k and iterate by replacing z with z^2 + c. If the iteration is bounded, the pair (c,k) belongs to the set. The Mandelbrot and Julia sets are just cross sections of this 4D fractal. If you know about Newton's method for solving polynomial equations, try applying it to an equation such as x^4 = 1. (In the complex numbers, this has 4 solutions: 1, -1, i, and -i.) Starting with the complex number c as an initial estimate, color the point c according to which solution Newton's method finds. You'll have four patches of solid color near the four solutions, but the boundary between them will be fractal, like the boundary of the Mandelbrot set. There are many other fractals that look little like the Mandelbrot set, such as the Cantor set, the Sierpinski gasket, and the Koch snowflake. (These fractals have nothing to do with complex numbers.) I don't have the space to describe all these now; if you're interested, you should be able to find them in some of the many popular books on chaos or fractals. James Gleick's book _Chaos_ (Viking, 1987) is widely cited. As you might guess from the last paragraph, chaos and fractals definitely have a lot to do with each other. Near the boundary of the Mandelbrot set, a small change from one point to the next can make a big difference in the point's membership in the set, and there's no way to predict this change without going through the entire calculation. That's the hallmark of chaos: sensitive dependence on initial conditions. Basically, any graph of a chaotic process is likely to be a fractal, and every fractal is the graph of some chaotic process. -Doctor Toby, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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