The Mandelbrot Set

Benoit Mandelbrot investigated the mathematical properties of fractals and discovered the central fractal form now named after him as the Mandelbrot Set.

The Mandelbrot Set is calculated using a process of iteration. Where we would be more accustomed to seeing an "equals sign" ( = ) in a mathematical equation, the Mandelbrot Set uses an “iterates to” symbol as shown above. If our mathematical statement says “a iterates to a+c”, then a becomes a+c. The (a+c) then becomes the new a. Next, the new a is run through the process again, so the next value of a, which was a+c becomes (a+c)+c. In the next iteration [(a+c)+c] becomes [(a+c)+c]+c.  The result then just keeps being looped back going round and round so what started as “a” becomes a long string with an extra c being added for each iteration.

So, to recap:

a iterates to become a+c,
a now becomes a+c,
a which is now (a+c) iterates again to become ((a+c)+c)
a which is now ((a+c)+c) iterates again to become ((a+c)+c)+c
a which is now ((a+c)+c)+c iterates again to become (((a+c)+c)+c)+c
and so on.....

In theory this lopping can continue forever. If say a=1 and c=2, we see the iterations would go 1,3,5,7,9,11.... We can see that this iteration would keep increasing until it reaches infinity.      

The statement that generates the Mandelbrot Set, has the surprisingly simple formula of z iterates to z²+c (i.e. (z multiplied by z) + c). The c in this case is a complex number, meaning it is a multiple of the number "i", which is the square root of -1. A graph is created with real numbers along one axis and complex numbers along another. Every point on the plane that is created is then iterated according to the formula z iterates to z²+c.

When we take any number and multiply it by itself time and time again (eg z multiplied by z, multiplied by z, multiplied by z, multiplied by z) one of two things will happen. If z is greater than 1, the resulting number will get bigger and bigger and bigger until it becomes infinite. If z is a number between 0 and 1, the result will get smaller and smaller approaching closer to zero the more times it is iterated. (Eg. 0.5 x 0.5 x 0.5 = 0.125; x 0.5 x 0.5 x 0.5 = 0.015625). When graphed, points on the Mandelbrot Set that tend towards zero as they are iterated are coloured black and points that tend towards infinity as they are iterated are assigned a colour depending on how fast the number approaches infinity.

When we actually come to plot the Mandelbrot Set, we find not the dull graphs you may remember from school but the most amazing, beautiful, organic looking pictures you can imagine as seen in the accompanying diagram.

The Mandelbrot Set is a fractal and is therefore self similar. That is, patterns are repeated again and again as you move in to different levels of magnification. The next diagram shows the basic pattern of the whole set repeated almost perfectly at a smaller level. These little shapes are repeated in all sorts of places all over the whole set. Fractals in nature repeat their pattern a few times and then the scale becomes too small to keep going, and computer based programmes reach the end of their ability to undertake the enormous calculations required. The Mandelbrot Set, as a mathematical formula, can repeat endlessly to infinity irrespective of how many times the image is magnified.

When we magnify portions of the Mandelbrot Set we find the most marvelous, organic, dynamic pictures that feel much more like art than mathematics.