Complex Numbers in Fractal Algorithms

Question

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?

Answer 1

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.

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:

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

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