Quote:
Originally Posted by Mechphisto
OK, I can grasp the idea of space as a fabric. The analogy of a mass being like a weight on a trampoline, taken to 3D. I can get that.
I can't get this geometry of the universe thing. I mean, I can imagine the universe being a ball or toroid.... but flat? Flat, beyond 2D, implied a thickness, which implies a 3D object. (Likewise saddle-shaped.)
So in all these cases, there's an outer edge. Meaning, if one travel long enough you'll reach the edge. I don't buy that, and since I'm not an educated astrophysicist, the only answer is I have it wrong.
OK, parallel lines forever. I get that concept on the surface of a sphere. But, and here's the crux of why I don't get this geometry topic--we're not on the surface of an object! We're, in a manner of speaking, INSIDE the object. So, if I were to imagine myself inside a sphere, and shoot two parallel lines, yeah, they'll remain parallel...until the reach the edge. Likewise if I were some raisin in the middle of a bagel, the lines would reach an edge.
What then?
Does this make any sense? (Sometimes naive questions from ignorant people can be just as baffling as educated answers from smart people.)
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You could try thinking of it like this:
Imagine 3d space. (Forget time for now and don't ask me how to visualise 4d spacetime - I haven't got a clue!
)
So you have your 3d space - you could be thinking of a cube, right, then divide it up in to a three dimensional grid so it looks like a stack of building blocks, or if you rather, like a 3d wire grid.
Then just subtract one dimension. (I do it by taking away the building blocks and imagining the 'impression' they would have left on a sheet of paper, or rubber.)
You end up with the rubber sheet 'grid with the bowling ball' familiar from the illustrations we see in books and on the 'net. And it isn't invalid because what happens in three dimensions also happens in two - they just take one away to make it easier to 'draw'.
And then... the universe is 'flat' in the sense that the rubber sheet is flat, and the missing third dimension is
also flat. (And of course the alternative geometries of the universe have the rubber sheet in the shape of saddles, spheres, donuts, soccer balls, etc... but in each visualisation the third dimension is 'missing'.)
BTW I've seen 3d illustrations of the bowling ball scenario as well: if you imagine your 3d grid, when you add the bowling ball and everything curves due to gravity, your '3d wire grid' becomes pinched in around the bowling ball like the waist of a mannequin.
Hope that helped? That's how I do it anyways...