Types of Fractals

Mandelbrot Sets

WHAT ARE THEY?

Mandelbrot Set is a type of fractal formed by formula iteration. Its concept is closely linked to the concept of a Julia Set. A Mandelbrot Set is formed using very similar algorithms as the Julia Set. To draw a Mandelbrot Set, do the following:

1. Make a complex number z equal to 0 + 0i
2. Choose some point on the complex plane, and make a complex number c equal to its coordinates.
3. Make z = z2 + c and repeat this change a lot of times.
4. If the number did not go to infinity, it belongs to the set and you can mark it. Otherwise, you can color the point depending on how fast it escaped to infinity.
5. Repeat steps 1-5 for all the points on the plane.

Notice that the algorithm is the same as for the Julia Sets, only the Julia Set algorithm varies z and keeps c as a constant, while the Mandelbrot Set algorithm varies c and uses z as a starting point (0, 0). For every formula, there is a Mandelbrot Set and an infinite number of Julia Sets. For the most commonly used formula z = z2 + c the fractal formed is perhaps the most famous fractal:

Other formulas can be used as well. Interesting fractals can be produced by trigonometric or logarithmic formulas:

 z = Arccos(z) + log(c)

Interestingly, the way a Julia Set looks depends on whether its constant c is located on the Mandelbrot Set with a corresponding formula. If the constant is located on the set, the Julia Set will be a completely interconnected figure, while it will be a set of disconnected points otherwise.