The Mandelbrot Set

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Please note that you should have heard of complex numbers in order to understand this article.

"To gaze upon geometry the way it is usually presented is a good way to turn a young mind to stone."1

This entry is an example for the other way of presentation, introducing one of the most common elements of fractal mathematics - the Mandelbrot set.

This set is named after its discoverer BenoƮt Mandelbrot who was a French-Polish mathematician living from 1924 until 2010. It is defined as the set of complex numbers c for which an=an-12+c with a0=0 stays convergent as n tends to infinity.

Huh, what?

Look at the formula. It describes a recursive sequence starting with 0 and repeatedly adding c to the square of the previous element. The first element is very simple to compute: With a0=0 a1 has to be 02+c=c. After that, the abstract representation becomes gradually more complicated. Thus we conclude with a glance at a2 in order to clarify the formula a little more: a2=a12+c=c2+c. The process applying the same formula again and again is called iterating, with an iteration being a certain element of the generated sequence.

What we are interested in is the sequence's behaviour starting with different values of c. For a real 2c we will easily see that the sequence converges for any c greater than -2 and less than or equal 0.25. For c= -2, the sequence will remain constant3.
For any other c the sequence will rapidly grow above all imaginable borders.

That alone is not interesting enough to justify its entry. But we did not yet consider c to be a complex number, and this is where chaos arises. Try a few complex cs and you will see that the sequence's behaviour is completely unpredictable after a few steps.

First we need to distinguish diverging and converging sequences. We do this by regarding the respective absolute value of each element. As soon as this value exceeds 2, the sequence of any complex c will diverge, so this is our limit.

Now we come back to the fun part. Remember the definition of the Mandelbrot set: It is the set of all c for which the sequence defined above will not diverge. At this point the set is nothing more than an abstract quantity of numbers, but this abstract quantity can be drawn.

Have a look at the complex plane. If we mark all numbers (i.e. points) that belong to the Mandelbrot set, we get an interesting form that looks like a pair of buttocks with antennas rotated 90 degrees anticlockwise.

Time for some aesthetics. Pick a nice colour gradient and paint the plane in the first tint. Then look at all points whose absolute value exceeds 2 and paint them in the next colour - you should have a perfect circle by now with a radius of two units.

Now apply the formula above to the points left and paint those who do not diverge in the next colour. You should have got the idea by now. The more often you repeat the last step the more the still converging set of points approaches the perfect Mandelbrot set. After a few thousand iterations, the result should look like this. Note that only the black area belongs to the set.

Even very pronounced mathematophobes will admit that this looks very beautiful, but maximum awesomeness is not reached until you zoom inside the shape.

As any infinitely iterated fractal the Mandelbrot set has a circumference of infinite length. Therefore its border shows manifold fascinating structures as you can see here, here or here just to show the possible diversity of forms. The last example shows a very common shape, namely a smaller copy of the complete set.

The self-similar satellites show two interesting properties. First, although there is an infinite number of them4 no two copies are rotated in exactly the same angle5. And second the whole Mandelbrot set, including those satellites, is connected - there is not a single isolated subset. Even the satellites which look as isolated as a set of points can have small antennas leading to the main area6. Also, the Mandelbrot set is full, which means there are no holes in it.

Looks nice, but what is it good for? Er - It looks good. Basically, that's it. The Mandelbrot set has no direct practical applications, but mathematicians appreciate it anyway (or because of that). It had a huge impact on mathematical thinking, pushed fractal geometry forward and lead to a completely new view on several further fields, especially chaos theory. For physicians believing in the existence and usefulness of a theory of everything it has similar consequences like Langton's Ant. After all it is spectacular enough to be allowed not to have a proper use.

Pictures by Wikimedia - They just have all the good ones.

1Leonard Mlodinow (American physician, *1954)2For those who did not read the disclaimer on top of this entry: Real numbers are all those you can imagine. Complex numbers are the other ones.3With c=-2=a1, a2=(-2)2-2=2 and an=22-2=2 for any n>3.4There is not much in the Mandelbrot set or in fractal geometry at all that is not infinite.5Apart from those crossing the abscissa. As you may have guessed, there is also an infinite number of these...6Most of these antennas are infinitely thin, by the way, although further structures are bond to them.

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