Quote:
Originally Posted by Frog march
If you have two points on a 2-sphere, and draw a line between them, you can define points inside the 2-sphere, in 3D space.
I was just wondering, if the Universe were a 3-sphere, could you define points in 4dimensions, by using two points in our space?
Is this done?
maybe this is trite, but I thought I would post anyway.
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Yes. Yoiu can realize the n-sphere as the surface of an (n+1)-ball. and you can extend the idea of spherical coordinates to higher dimensions.
A 2-sphere is naturally realized as the points of norm 1 in 3-space -- the surface of the unit ball.
The 3-sphere is just the points of norm 1 in 4-space -- again the surface of the unit ball in 4-space.
This idea extends to the n-sphere in (n+1)-space.
While it is not in general possible to embed an arbitrary n-manifold in (n+1)-space, you can always do it with the n-sphere. You can also embed an n-manifold in a Euclidean space of suitably high dimension (Whitney embedding in dimension 2n). You can even do this isometrically for Riemannian manifolds, with a bit more work (Nash embedding). The theorem also extends to metrics with arbitrary signature.