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zhermes
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New thread continuing a topic raised in https://www.physicsforums.com/newreply.php?do=newreply&p=3865854
Okay, so I saw on wikipedia that you can have a 'flat torus' --- at least a 2 dimensional torus inbedded in 4-space. So you can also have a 3-torus, that's flat?
And would time just be added on as an additional, orthogonal dimension?
I'm a little confused about how/why this is considered flat. In the same wikipedia article, it says that a (2)cylinder is also flat---this is news to me. The analogy they make is that bending a flat piece of paper into a cylinder doesn't require any stretching/deformation of the paper. Okay. And I also realize that a 2cylinder would still have triangles whose angles add to 180 degrees... etc etc.
These things definitely aren't true for the standard 2-torus in 3D; I would have assumed the 3-torus was the same.
My understanding of differential geometry is rudimentary--only what I've gleamed from attempts at GR. I have no experience with 'topology' per se. None-the-less, equations would be welcome.
Nabeshin said:The 3-Torus is actually spatially flat everywhere, so the measurements of flatness only support such a theory. One idea of how to place limits are whether or not we see radiation running along the compactified dimensions of the torus, i.e. multiple images of the same objects. This of course gives you only an lower limit to the 'radius' of the torus, but I don't know of any actual experimental bounds on this from CMB data, for example.zhermes said:How does this interplay with measurements of flatness? [...]Nabeshin said:Yes, the simplest of them being a 3-torus.neginf said:Could physical space have a different topology than the usual R^3 ?
Okay, so I saw on wikipedia that you can have a 'flat torus' --- at least a 2 dimensional torus inbedded in 4-space. So you can also have a 3-torus, that's flat?
And would time just be added on as an additional, orthogonal dimension?
I'm a little confused about how/why this is considered flat. In the same wikipedia article, it says that a (2)cylinder is also flat---this is news to me. The analogy they make is that bending a flat piece of paper into a cylinder doesn't require any stretching/deformation of the paper. Okay. And I also realize that a 2cylinder would still have triangles whose angles add to 180 degrees... etc etc.
These things definitely aren't true for the standard 2-torus in 3D; I would have assumed the 3-torus was the same.
My understanding of differential geometry is rudimentary--only what I've gleamed from attempts at GR. I have no experience with 'topology' per se. None-the-less, equations would be welcome.