There’s a difference between ‘dimension’ as used in a pulp sci-fi (‘Attack from the fifth dimension!’) as opposed to a scientific sense. In the former, the word is used roughly synonymous to ‘world’ or ‘universe’ – these dimensions are some place you can travel to. In the latter, dimensions are a characteristic of a space – roughly, they are more like directions you can travel in.
Take our world: you can travel forward (or backward), left (or right), or, with the right means of travel, up (or down), making for three (spatial) dimensions. On a sheet of paper, you can travel only up/down, or left/right, without leaving it – it’s two dimensional. A single line has only one direction to go in, on a point, there’s no direction at all, so these ‘spaces’ have a dimension of one and zero, respectively.
This also means that you can specify each point in some space by 1) specifying a point of origin, and 2) specifying how far to go along each of the dimensions in order to get there. So, on a piece of paper, you might start at the lower left edge, and give the location of any point on the paper by first giving the distance you need to travel to the right, and then giving the distance you have to travel up in order to get to that point. These two numbers then identify any given point on the paper – they give it a name, its coordinates. This is then another, equivalent, way of specifying dimension: the number of numbers needed to identify any point in a space is its dimension. On a line, you only need to specify how far to travel – one number, one dimension. On a point, well, there is only one point, which is already identified – no number, zero dimension. In everyday space, you need to specify how far to travel up/down, left/right, forwards/backwards – three numbers, three dimensions; this is the same as when you have a rectangular package, and its measurements are given as its width, length, and height.
From this point, ‘higher’ dimension are not such a strange step: you could simply consider a ‘space’ in which you need four numbers to uniquely identify each point (technically, our space is such a space, however, the fourth number is a temporal, rather than a spatial, coordinate: if you wish to schedule a date, giving just the location won’t suffice to specify the meeting point uniquely, you’ll also have to give a time; however, for the moment, let’s contend ourselves with spatial coordinates). As far as a ‘direction to travel in’, this fourth coordinate is rather hard to visualize – it would be a direction ‘at a right angle to’ all of the ordinary directions of our three dimensional space, much as the direction forward/backward is at a right angle to the two directions on the paper plane. However, as much as this third direction does not ‘fit into’ the plane, the fourth direction does not ‘fit into’ familiar three dimensional space; and since our imagination is three dimensional – by habit, if nothing else --, we can’t readily form a picture of such a direction.
But mathematically, there is no obstacle! You can talk meaningfully about four, five, six… even infinitely many dimensions, and in fact, many mathematicians do this quite routinely.
The question then is, as for physics, which is supposed to describe the world we observe – why would you want to do that? Well, it turns out, that some theories, in order to be consistent, need a higher number of dimensions – like, for instance, string theory (though to what extent it describes the world we observe, if at all, is rather controversial). So, the theory, involving in most formulations no less than ten dimensions, is easily (more or less) written down – but then, what to do with all the surplus dimensions? If the world ‘actually’ has more (spatial) dimensions than the three we observe – why don’t we see them? Seems like a hell of a thing to just never have noticed!
So one has to figure out a way to ‘hide’ these extra dimensions, and that’s where compactification comes in. Basically, what you do is something analogous to taking your two-dimensional piece of paper, and roll it up into a cylinder, as tight as you can. Now, this is still the same two-dimensional piece of paper, but, from a distance, you might not easily notice the second dimension – it might look quite like a line, like a one-dimensional space. You have, at least on sufficiently great distances, effectively gotten rid of the second dimension.
So that’s, in a pinch, how you (hope to) extract our three dimensional world from higher dimensional string theory spaces – we’re just big enough such that we don’t notice the extra dimensions! That said, they should, if we look hard enough, still have some observable effects – we’d just need a magnifying glass big enough, which we’ve build, at least for relatively large extra dimensions, with the LHC at CERN. Unfortunately, all it’s done so far is to report back that there’s nothing to report (which of course is a very valuable result in itself), i.e. that if there are any extra dimensions, they must be smaller than a certain size, or else, we would have seen them by now.
