My guess is that only three of those figures 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). Actually, we can't even imagine what it looks like since we live in 3D, but all of the lines you see in that figure have the same length and are straight in 4 dimensions. Don't think about it too much or you'll get a headache.

SIDENOTE: Hypercubes (ie n-dimensional cubes) have some interesting properties related to graph theory. Supercomputers are often networked in a hypercube pattern - each corner represents a computer and each line represents a network connection. This is done so that computers can pass information to each other the fastest way possible with the least amount of connections.

Notice that in order to get from one dimension to the next, you take two of those figures and connect all the analagous "corners" with more figures of the same size. So if you take two lines and connect their endpoints with lines, you get a square. Take two squares and connect each side with more squares and you get a cube. That's how you can figure out what a 4D hypercube is like.

Now that we have a idea of what whole-dimensional figures look like, and how we jump from one dimension to another, let's take a look at another essential characteristic of geometric figures: scale

Relative Scale

When I was a kid, sitting bored in church, I used to look at someone far away and pretend to squash their head with my fingers. It was a silly fascination since their head could obviously never fit between my fingers, but since I was far away the "scale" of their head became small.

Here are three examples of objects with their scale doubled

This figure is almost embarassingly simple, but there are a couple of important subtleties to note. Notice how the only factor that affects the scale in the square and cube is the length of their sides. So you can "stretch" each line to whatever scale you want and that's what determines the scale. Another difference I would like to note here is that volume/area is different than scale. When I say that one square is "twice as big" as another square, I mean that the scale from one to the other is 2 - NOT that one square has twice the area as the other.

Calculating Euclidean Dimension

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

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. What this section will goes over is how to actually calculate the dimension of a line, square and cube.

The key to calculating dimension here is using the self-similarity. Notice how a line can be drawn by putting a few lines together, or a square by drawing a bunch of little squares. If we can figure out how scale fits into that idea, then that's dimension. Take a look at a square: you can divide it up lots of different ways: 4, 9, 16, 25...

You can just as easily do this with a line or a cube. Take a look at the cube's dimensional calculation.

The pattern is pretty clear here: the dimension is just the exponent. The figure below showsthe formula for dimension. In case you aren't familiar with the "log" function, it's a way to calculate the exact number of the exponent, that way we can put the variable d on its own side of the equation.

 

Calculating Fractional Dimension

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

The Van Koch curve: this curve has an infinite length but the area under this curve is finite.

Let's take a look at how this figure is constructed. Here is how each step is done, ad inifitum.

The scale of each small piece to a large piece to a large piece is 3, and at each step you use 4 figures. Below is the calculation for the dimension of Van Koch curve.

1.262 is an approximation to four significant figures

Comment on constructing the Van Koch Curve: do you notice any similarities between how the above fractal is constructed and how a square is constructed from two lines? This idea of taking one piece and connecting it to another with other pieces is what leads this "self-similar" characteristic of fractals (really self-similarity is related to dimension, it's just easier to see with fractals).

Now this is only an introduction to how dimension is calculated in very simple fractals. There are other fractals where it's not obvious how to calculate their dimension (eg The Mandelbrot Set, Julia Set). I hope that this information has been helpful in your understanding of how dimension works and what a fractal really is.

Page last updated: February 04, 2008