One of the most common questions I get from undergrads is What’s the deal with string theory having <blah> dimensions?; more often than not, they have no idea how many dimensions there are supposed to be, so they just say some number larger than 4. This is a more fun question than the – even more common – question, Is this going to be on the exam?!

I know very little about string theory itself, but I’m usually able to explain to them what the hell people mean when they talk about the dimensions of string theory – today, you learn this too. An early incarnation of string theory required 26 dimensions, however when people speak of string theory now they are referring to a 10 dimensional theory; then there’s M-theory, which is 11 dimensional! But what the crap does this even mean!?

Previously I’ve discussed time as the fourth dimension, however for the sake of simplicity let’s just think about our regular 3-dimensions that we’re used to. In order to imagine adding extra dimensions, we should first consider one and two dimensional things in comparison to our usual 3 dimensions. Imagine a stretched out piece of fishing wire as your one dimensional example, and a piece of paper as your two dimensional one; obviously any regular old solid object is a 3 dimensional one.

If you imagine the fishing wire stretched between two walls, you could describe a point on the wire in terms of one number – the distance from one of the walls. Now zoom right in on the fishing wire! If you look closely enough at it, the wire is actually a REALLY thin cylinder; an amoeba on the wire can move along the wire, but it could also move around the “cylinder” around a big circle and back to where it started. For the sake of analogy, let’s call the direction along the wire as a macroscopic dimension and the freedom to move around the wire/cylinder is a curled dimension.

Like these amoebae, we think of the strings of string theory as being small enough to see (and move through) some similar notion of curled dimensions. The cylinder above was an example of having a one dimensional line with a circle attached at every point on the line; think about it for a minute. Any cylinder is a straight line with a circle attached to every single point – infinitely many of them. While it’s not nearly as simple to picture, conceptually you can imagine attaching a tiny circle to every point in space – so small that we can’t possible see/notice them! But such that objects far far smaller than our regular subatomic particles can move through; the strings of string theory can move through extra dimensions like this!

You might be surprised to know that this notion of extra curled up dimensions has been around for almost a century now! In fact Kaluza and Klein proposed a model of uniting the gravitational and electromagnetic by EXACTLY adding a circle to every point in space-time. Very loosely imagine the electromagnetic field as a collection of sin waves,

Just grabbed from Wikipedia, don’t worry about the labels.

so that they can be “shifted” along by a fixed amount and will appear to be unchanged, like pac-man walking along the screen and getting back to where he started; or like moving around a circle! So there is a very nice (albeit, ultimately incorrect) geometric theory incorporating the electromagic field through adding a circular curled dimension!

String theory postulates that everything is made up of tiny strings oscillating in all our regular dimensions and six extra curled dimensions. And all the different particles are just the strings vibrating in different ways! I honestly don’t know anything about string theory, but I think it’s a very nice idea; the universe is a symphony.

And you thought it was hard to draw 3D perspective – this is the six dimensional object that’s attached to every point in space-time – a Calibi-Yau manifold.