What Andy is Excited About

Remember your high school geometry? (Don’t worry, there won’t be quiz.) Remember dimensions? A 1D figure is a line, 2D is a plane, and 3D is space. Well it turns out that there are more than just those 3 dimensions…. that’s what what we call a fractal!

What are fractals? Fractals are figures with fractional dimension that often exhibit self-similar properties. In this article, we’ll briefly cover how fractional dimension actually works.

Euclid's view of geometry, with cubes in dimensions 1 through 4.

Before jumping into fractional dimensions, let’s look at what whole dimensions look like. My guess is that only three of those figures to the right are familiar to you. The hypercube is something that exists in 4D, so drawing it in two dimensions is rather difficult (hence the confusing lines that cross each other). Don’t think about it too much or you’ll get a headache – we’ll talk about hypercubes another time.

Notice that, in going from one dimension to the next, you take two figures of the smaller dimension and connect all the analagous “corners” . So if you take two lines and connect their endpoints, you get a square. Take two squares and connect each side with more squares and you get a cube.

Ok, so you can see how we construct an N-dimensional figure from figures with a lower dimension.

So far it seems like Euclid arbitrarily assigned numbers to each dimension; it seems like there’s no reason why he said that lines had a dimension of 1 and not 4.352. Turns out the man had a reason. Let’s take a look at the dimension of a square.

Notice that the exponent in both cases is 2, so we have a 2D figure

For this example, let’s exploit self-similarity by splitting a square evenly into smaller squares. You can divide your square up lots of different ways: 4, 9, 16, 25, etc.  You might notice, though, that no matter how you divide your square, the number of squares you need is the number of squares you need for the line raised to the power of 2.  (this is why saying “raising to the power of 2″ is the same as “squared”).

Huh? The dimension of the figure is the exponent when you construct your figure from a figure from smaller dimension. So if we need 3 chunks to construct a line, we need 9 chunks to construct our square. For a cube, you would need x^3 small cubes to make a big cube.

So how do we use what we know to calculate the fractional dimension of this snowflake figure?

The Van Koch curve

Let’s take a look at how this figure is constructed. It’s constructed iteratively, so we repeat the following steps to infinity.

The scale of each small piece to a large piece to a large piece is 3, and at each step you use 4 figures. So, even though the size is 1/3 the size, you need four figures to construct the larger figure. Below is the calculation for the dimension of Van Koch curve.

Where S is the size of the line, N is the number of figures needed to construct the whole figure and d is the dimension we are calculating.

So our Van Koch figure is somewhere between 1-dimensional and 2-dimensional. Also, figures like this have weird mathematical properties, like having infinite surface area and finite volume.

Cool, huh?