The Mandelbrot set

The Mandelbrot set is named after Benoit Mandelbrot. He is a Polish – born, French and American mathematician.

He is largely responsible for the present interest in fractal geometry.

From his dedicated study of fractals came the great Mandelbrot set.

Fractals are never – ending, infinitely complex repeating patterns. The Mandelbrot set is a collection of numbers that are different from the real numbers you see in your every day life. These all patterns are happening in the set of complex numbers.

 

Every complex number has two parts- the real part and the imaginary part. Every complex number can be written as

$ z = a + \imath b$
Where the number i is the imaginary unit and $\imath_2 = – 1$.

This notation is very convenient because from this form we can easily find this numbers place in a complex plane.

So, how do we get to Mandelbrot set from here?

 

Let’s take some complex number b and let’s associate with it the following function.

$ f_b(z) = z^2 + b$
We are interested in the behavior of our function in zero and then in that result and then in the next result and so on.

Now we’re observing how this function will behave if we take $ b = 1$.

$f_1(0) = 0^2 + 1 = 1$
The function value in zero is equal to 1, so we’ll observe how it acts in 1 and continue the process with the following results.

$ f_1(1) = 1^2 + 1 = 2$$ f_1(2) = 2^2 + 1 = 5$

$ f_1(5) = 5^2 + 1 = 26$

…

The Mandelbrot set observes what happens to the size of these numbers. This size is the distance of the given number in a complex plane and the origin.

There are two options. The first option is that the distance from zero of the sequence gets very, very large which means it blows up – the size of the number goes to the infinity. The second option is that the distance is bounded- never gets larger than two.

Let’s observe the process we did. We got numbers $ 1, 2, 5, 26$ which means they are getting larger and larger and will continue to do so. This means that we’re talking about the first option.

Now let’s take a look what happens when $ b = -1$. We’ll always be getting numbers zero or $ -1$. This means that this is the second case.

Mandelbrot set is the set of complex numbers in which case two occur.

So how can you draw all these patterns? You just start iterating zero under $ z_2 + c$, if it takes a long time to get bit, you give it one color, and if it gets big very quickly you give it other color. This is interesting because you can’t predict how your function will act if you move change c just a little bit.

If we take fixed value of c, and continue observing this polynomial as we observed ones in Mandelbrot set we’ll get something called the Filled Julia set.

This means that the Filled Julia set is the set of complex numbers z so that under iteration by fc the values don’t blow up.