Dimensional Discourse

Projecting from n dimensions to m dimensions is a relatively simple process. My favourite doodle since childhood is the classic hypercube projection onto 2d. Visualising simple 4+d objects, such as the higher-d versions of the Platonic solids (the regular polytopes) isn't too bad (once you've spent several years working with them), but in general one uses projections to understand and visualise.However, it may surprise you to realise that you haven't even begun to visualise 3d spaces! Take the surafce of the Earth, a 2-sphere as we call it... it's a surface and hence 2d. But try and visualise it without using 3d space into which it can be embedded? You can't. This doesn't mean that a 2-sphere is 3d.Now imagine the 3-sphere. It's a 3d space so imagine yourself inside it. You can float around, but if you float far enough in any 3d direction you eventually get back to where you started. So what does this space look like? To visualise it, you need to imagine it embedded in 4 dimensions. But the space itself is still only 3d.This is why sci-fi wormholes (Deep Space 9, Farscape, etc) are always depicted completely incorrectly. The artists have looked at our drawings and used them directly without realising that they are dimensional reductions because a wormhole is a 3d object that can't be simply visualised. Winds me up no end

Exactly! Though the Asteroids playing field is not a 2-sphere, it is a 2-torus. If you wrap it up to visualise the space you get a doughnut. Welcome to topology!How do you know it's a 2-torus? Wrap a loop of string around a 2-sphere (surafce of the earth, a ball, etc) in any way you like, and you can always shrink it down to a point. Now take one end of your string and move off the left edge of the screen, and it appears on the right. Tie it into the loop. Now try and shrink your loop... you can't, it's stuck on the 2-torus. You can do the same in the vertical direction as well. In fact, you can count how many wraps you make in both directions, call them A and B. The 2-torus can be identified by the algebra of A+B. Welcome to algebraic topology!You can have a 3-torus, where we live in a space that repeats itself in the 3 orthogonal directions; a bit like the 3-sphere but different for similar reasons as given above. My first research, done as an undergrad back inthe 80s, was into investigating whether our universe exhibited this kind of behaviour.

Well, yes and no Curvature of space and time is a local property, and is what we call geometry. We can see its effect all around us (we "fall" to the ground) The large scale property that we are mentioning here cannot be so easily observed. It is global and is what we call topology.And you don't need one for the other. Crashfrog's asteroids game has a completely flat space and yet it still wraps around on itself. Any curvature here is purely an effect of visualising the space (as in wrapping it up to make a dougnut) and isn't real. We call this extrinsic curvature. The curvature of space-time that leads to you falling to the ground is very real and is intrinsic curavture.This message has been edited by cavediver, 03-22-2006 10:07 AM