Philosophy Of The Cosmos - Dimensions
You guys are fantastic at anticipating what's coming later in the syllabus! One of the things which keeps cropping up in your discussions is the problem of "visualising" more than 3 dimensions.
I'm at a bit of a handicap here because I'm not sure quite what you're after when you talk about "visualising" higher dimensions. But here's an idea that I think will help at least a bit. And we'll have more about this in lectures later.
The most important thing to remember about a dimension is that it's _only_ a way of measuring things. Something has X dimensions if we use X numbers to describe it: that's all "dimension" means.
In other words, it's just maths. As Daniel says over on Philosophy Of The Cosmos - The, "Maybe looking at the maths behind such a figure would make it feel a little more graspable as a concept." Only that's an understatement. The maths is not just a way of making something more graspable. When it comes to dimensions, the maths is all there is.
For example, if we want to know about vegetables, we might be interested in their colour, their cost, the amount of water per kilogram, the amount of vitamin C per kilogram, whether they taste nice to me, and whether they taste nice to my alpacas. That means that vegetables live in a 6-dimensional space. This is not a different, metaphorical way of using the word "dimension": it's exactly the same as the way it's used in cosmology.
In current physics, using relativity theory, we use 3 spatial dimensions and 1 time dimension. As you will see in lectures next week, a time dimension is very similar to a spatial dimension but not exactly the same. ALL THIS MEANS IS THAT WE MEASURE THE LOCATION OF A POINT IN SPACETIME USING FOUR NUMBERS: three to give its spatial location in a frame of reference and one to give its time location in the same frame of reference. Visualising what this _looks_ like is easy, on smallish scales, anyway: locally, it's just how we've always looked at the world, with the maths being different only when objects are moving very fast relative to our frame of reference, or when they're highly charged or very massive.
We could turn our 6-dimensional vegetable space into a 7-dimensional space, by adding a number representing something else, like say whether they turn tasteless when cooked. And similarly we can do cosmology with more than 4 dimensions if that turns out to pay dividends.
(And here's an interesting side issue: There doesn't even have to be a right answer about how many dimensions there are in physics any more than there's a right answer for vegetables. It could be that for some purposes it's good to count 3 dimensions, for some purposes it's good to count 4 and for some purposes 9 or 10. It MIGHT turn out that there's only one way of describing the location of a point in spacetime, and that it requires 4 or 9 or 10 numbers. IF that happens then we'll say that spacetime has that many dimensions. But we're not there yet — we don't know whether this is going to happen. All we know so far is that it's useful to use at least 4 dimensions.)