All Shapes and Sizes

Let's try not to be so linear. We tend to think in a very Euclidean sense (and why shouldn't we?) when we think about measuring space and objects – and not only that, we also tend to think rather continuously, as opposed to discretely. However, measuring things can actually be done in a variety of ways. Let me explain what I mean.

First of all, I'm currently studying for a measure theory final, which might explain why any of these things are in my head in the first place.

What is measure theory anyway? Basically, a lot of measure theory deals with the measurement of objects – some of which we already how to measure. I can ask you to find the length of a line (or an interval, like [1,5]) or the area of a square or even the surface of a sphere (though I recently completely forgot the formula for exactly that calculation at a pub quiz a few weeks ago – quite embarrassing). If you've had some calculus, you also know how to measure the area under a curve by using an integral. And we also know how to calculate the volume of things like spheres, cubes, or other shapes.

As you might have noticed (though we don't always say it so directly) all of these measurements have an inherent dimension. 'Length' is always one-dimensional, 'area' two-dimensional etc.  Obviously. We just say 'length of a line' instead of saying 'the default one-dimensional measure of a line' because, since it is the default, we don't think that there could be other ways to measure such a thing. Measure theory looks at some of these other ways. But I'm getting a little ahead of myself.

I mentioned earlier that we also tend to think continuously as opposed to discretely. Things that are discrete are a little bit different. For example, if we consider the interval [0,1], which includes all numbers between zero and one, that's a continuous chunk of numbers. If you were to draw it, intuitively, you would not pick up your pencil or pen – you would just trace along a number line from zero to one, hitting all the elements in that interval. However, if we think about the set {0,1} (notation is very important in math – [0,1], (0,1) and {0,1} all mean different things!) – then, the only things in that set are the number zero and the number one, nothing in between. Two unconnected dots on a number line. A discrete set.

And what measure theory tries to do is to measure things like these sets. But imagine trying to use that intuitive 'length' notion to measure something that is discrete – something whose elements are separated by space, even if it's a very small amount of space. How does that even work? Does it even make sense to try to apply that definition? Not really.

We need a new idea of measurement.

So, that's some of the fun stuff we've been getting up to in the lecture this semester. If you'll let me use as a premise that we do have a tool (several, in fact) that can handle discrete inputs and measure them – then we can come back to the idea of dimension. What if you're trotting down a road in the mathematical world and you come upon a set Ω. Very startled, you want to know some things about it – its measure, for instance. But we know nothing about its contents. They could be related to a function or an algorithm. They could be points, vectors, or even  – gasp! – nxn-dimensional matrices. We really don't know much. So, how do we find out what dimension it has? 

What are some things that could go wrong here? I claim there are several ways to get the wrong impression. What if this set you happen to meet is actually just a kind little square, but you don't know that, and mistakenly, you try to measure it with a three-dimensional measure, like volume. What happens if you try to figure out the volume of a square? Well, you get zero. Just like if I tried to measure the length of a single point. If my object is n-dimensional and I try to measure it with a measure any bigger than n (in this example, n+1), then I get zero.

And what if I make a mistake the other way around. What if the set I encounter happens to actually be a cube and I try to calculate its area. I don't just mean the surface area – the surface is two-dimensional, that wouldn't give us any trouble. But the area of the entire cube? That would have to be infinite.

So, let's get back to that road in the mathematical world and back to Ω. You want to figure out what its measure is – but in order to do that accurately, you have to figure out what dimension it has.

I'm skating over a few technical details here, but basically, in one area of measure theory, we talk about something called the Hausdorff measure. (Hausdorff was, as you may have guessed, some old math guy. He actually did quite a lot of cool things.) The definition of the Hausdorff measure is pretty hairy – it looks a bit like this:

(If I have any math critics reading this, I realize this is the delta-approximative, s-dimensional Hausdorff measure – I thought it was the prettiest formula to show here. The actual formula for the Hausdorff measure hides all the details by taking the limit.)

And just like with our intuitive standard ways of measuring things – like length and area and volume – there are different versions of this measure in different dimensions. And there is also something called the Hausdorff dimension. The Hausdorff dimension of a set is simply the dimension that is 'correct' – which is to say, if I try to measure that set with a dimension that is higher than the Hausdorff dimension, I'll get zero for the measure (like measuring a square's volume) and if I measure it with one that is lower than the Hausdorff dimension, I get infinity (like the area of a cube). So, to figure out what dimension that set has that you see walking towards you on a lonely road, simply measure it with a few different dimensions and see where the jump occurs. That's it's Hausdorff dimension.

Many of you will say: "But when will I ever be walking down a lonely mathematical road? When will I ever need any of this?" My answer: "Yes. But... isn't it cool?"