NOV

2009

Earlier this week, Slashdot published the news that a three-dimensional equivalent of the Mandelbrot set has been found. Visiting Daniel White's page, not only did I find an explanation, but also unprecedented images -- and I knew immediately that I would like to create such images myself.

### Fractured

The class of objects to which the Mandelbrot and the new *Mandelbulb* belong is called *fractals*. What are fractals, anyway? Well, a fractals in an object with a dimension that is not an integer. At first glance, it is not quite clear what this means: an object is either one-dimensional, such as a line; or a sheet of paper, which has two dimensions (ignoring its thickness); or, for example, a cube with three dimensions. But two and a half?

If we look at the dimension of a vector space, this is, indeed, impossible: there, one would simply count the (independent) directions that are to be found in a given object, and name there number the object's dimension: if there is just length (such as on a piece of string), the dimension is one. The living room has length, width, and height, thus it is three-dimensional.

But there is a different way: let us take a ball that is large enough to hide the piece of string. For a one-meter string, we would need a ball one meter in diameter. Then, try with a ball half that size: we would need two to cover the string. If we tried with ten-centimeter balls, we would need ten of them, and so on.

That does not work for my living-room, however: I could put the whole room into a large ball of, say, five meters; but I would need many more than just five balls of one meter each -- more than a hundred, in fact. That is, the dimension of an object tells us how quickly the number of balls grows when the balls themselves shrink. For a one-dimensional object, 2 balls of size 1/2 are sufficient, for something three-dimensional, we would need 8, or 2^{3}.

Now, let us turn around and use this counting as a definition: a chessboard is two-dimensional precisely because we need 64=8^{2} squares of size 1/8 to cover it.

### Erasing

What would an object need to look like to make a fractional number drop out of this computation? Let us make a little experiment. You only need a sheet of paper, a pencil, an eraser, and maybe a ruler (I did without). First, draw a straight line, like this:

Then, take the eraser to remove the middle third of the line:

Erase the middle third of the two remaining lines as well:

And so on -- in principle, you can continue indefinitely. That is, as long as your eraser is fine enough.

What happens if you try to cover your creation with balls? Start with one that is large enough to cover all those little dashes. If you then try balls a third of its size, you do not need three of them -- instead, two are sufficient: one for the left side, one for the right. You have removed everything in the middle with your eraser, after all. Using balls of diameter 1/3, only 2=3^{0.63} of them are enough to cover the drawing. So you could say your little piece of art has a dimension of 0.63. This is not as weird as it may look at first glance: There is an enormous number of dots, most of which are very close together. Isolated dots would have a dimension of 0; and dots that are infinitely close together form a line of dimension 1. Our result is somewhere inbetween, which should hardly come as a surprise.

The Mandelbrot set is similar: regardless of how far we magnify parts of it, there are ever finer structures appearing.In principle, this is true for the Mandelbulb above; however, the picture has been rendered with simplified equation, so only the largest knobs are visible: it is like throwing away the erasor after the first two steps.

### Appearing soon

During the next few days, I would like to show how to calculate the Mandelbrot set, and the new Mandelbulb. Hopefully, there will also be more pictures, and perhaps a movie or two.

[Edit: Typos]

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