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I've heard that our universe may be open or closed. If it's closed it might have a toroid shape. If this is the case, would that imply that our 3 spatial dimensions have to be embedded in a higher dimensional (spatially) space? For example, the old asteroids video game was apparently a torus mapped to a 2-dim video screen, and it stayed in 2 dimensions, but with a specific behavior of moving objects.

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It is rather arguable that non-trivial topologies imply that the space is embedded into higher dimensions. It is not sufficient to say that all the known toroids are embedded into 3d to prove that statement. – Alexey Bobrick Nov 21 '13 at 19:41
@AlexeyBobrick I'm trying to understand your answer. Did you say that all known toroids can be embedded in a 3-D space? A simple example is a 3-D doughnut in our 3-D space. With respect to the universe, I'm assuming it cannot be embedded in a larger 3-D space, like an actual doughnut. I wonder if the entire 3-D space could be a torus without there being a larger 3-D space or an n-D space (n>3) to contain it. – Mitchell Kaplan Nov 22 '13 at 1:11
I mean something rather simple, sorry for being confusing. What I meant was that there exist many 2d surfaces with non-trivial topology, which are embedded in our cartesian 3d space, including toroids (with manifold dimension 2). At the same time there are no physical examples of 2d objects with non-trivial topology, which are not embedded into our 3d. Now, my statement is that these reasons are not enough to extrapolate and say that "if spacetime is curved or has a non-trivial topology, then it necessarrily has to be embedded into n-d". – Alexey Bobrick Nov 22 '13 at 1:23
@AlexeyBobrick Thanks for the clarification. – Mitchell Kaplan Nov 22 '13 at 3:05
This feels like it would be much more on topic over on Physics.SE – Rory Alsop Nov 22 '13 at 9:17

1 Answer 1

The universe being open, closed, or flat only determines the type of geometry one must use to describe distances (and time). For open and closed geometries, Euclidean Geometry is not what one should use. I would also agree that our universe being open, closed, or flat has nothing to do with the number of dimensions it contains.

At present, there is no direct evidence that there are any additional dimensions over 3+1 dimensions (this just means three spatial dimensions and one time dimension). However, many GUT theories do include additional spatial dimensions in an attempt to unify all of the forces.

Also at present, we have very good evidence that the universe is 'flat'. What this means is that the angles of a triangle have to add up to $180^{\circ}$ and distances are measured in the standard Euclidean way. When we talk about the universe being flat, this is a purely global statement. Locally, however, it is completely possible to live in curved space. We actually do live in curved space. The mass of the Earth is curving space and time in a way that General Relativity predicts, and therefore clocks run very slightly differently depending on where you are on the Earth's surface, and distances are very closely approximated by Euclidean distances, though they're not.

How do we actually determine that the universe is globally flat, you may ask? We use what's known as a standard ruler. Much like a standard candle, if we think we understand the physics, any deviation from what we would predict gives us new information about the universe (in the case of supernovae, it's that they can be used to measure distances, and therefore map out the expansion of the universe). We use the angular size of fluctuations in the Cosmic Microwave Background Radiation (we think we understand the physics behind the fluctuations fairly well) to test how much, if at all, the universe deviates from flatness. The latest results from Planck show very good agreement with the standard picture provided to us by the standard LCDM cosmological model.

Below is the power spectrum of the temperature fluctuations of the CMB. The location of the first peak is what cosmologists use to measure flatness.


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