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I am a high school freshman who is undertaking a small development project on fractals. I do not want to get too in depth, but I would love to blow my math teacher's socks off. Having looked through various Wikipedia articles, I struggle to figure out how complex numbers play a role in fractal algorithms. Can anybody help me understand what exactly complex numbers do in the formation of fractals?

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I'm not sure I understand correctly, but there are some fractals (such as the Mandelbrot sets whose definition is easily formulated with complex numbers. I think they play a role because complex numbers can be used to do planar geometry. – Joel Cohen Sep 20 '12 at 14:33
In the case of the Mandelbrot fractal for example, the fractal literally is a subset of the complex numbers. I imagine the answer to what they "do" in the formation is that the formation of these kinds of fractals (pick which ones you want to look at) involve iterating an algebraic operation in the field of complex numbers and then performing some kind of check. (See JC's WP link above for more information.) – anon Sep 20 '12 at 14:34
Im no expert, but from what I see in the WP article, a Mandelbrot set would be quite hard for a high-schooler (who hasn't "officially" passed Geometry, but I have dabbled with derivatives ^^) to iterate... – fr00ty_l00ps Sep 20 '12 at 14:38
up vote 3 down vote accepted

To work with the complex plane born with the work of Fatou and Julia about rational functions, then the basic theory is on this numbers, specialy in the Riemann Sphere, but they cannot see this objets because they didn´t have computer. If we only consider the Real numbers for the iteration, f^n(x)=f(f(f(...f(x)))) n times, which is the principal idea behind fractals is relative simple, for example, for the function $x^2$ the points $x<|1|$ tend to $0$ lets paint in red, $x>|1|$ tends to infinity lets paint in blue and when $x=1$ 1 is a fixed point lets paint in black, the real line is painted in three colors and the fractal is not enough pretty. enter image description here

Now the natural way to generalize this is take a plane and associate to this the complex numbers then for the iterations we have points in the plane. The next step is take a region of the plane, for example, $[-2,2]\times[-2,2]$ and take a division of $100\times 100$ points, then we have 10000 pixels. Then we can associate complex numbers and pixels, like the functions work well with complex numbers we can proced to iterate. With the same argument for $z^2$ we have the next picture:

enter image description here

But if we take another function like $z^2+1$ we have:

enter image description here

The color are simple because black indicates points with simple behavior and green the Julia or fractal Points.

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