What is the Shape of the Universe?

OK, carrying on with my book on non-Euclidean geometry, this point comes up - what shape is the universe? I have done some googling and the below article is quite interesting. I was wondering what the current scientific/mathematical ideas are on this.

Quote:

Soon after general relativity was published, a number of theorists, including Einstein himself, delved for solutions that could describe the universe in general, not just the stars and other objects within it. The researchers discovered a plethora of diverse geometries and behaviours, each a distinct way of characterising the cosmos. Some of these models imagined space as resembling an unbounded plain or endless flat landscapes, only uniform in three directions, not just two. Two parallel straight lines, in such a spatial vista, would just keep going in the same direction indefinitely, like outback railroad tracks. Physicists call these flat cosmologies. Other solutions possess spaces that curve in a saddle shape, technically known as hyperbolic geometries with negative curvature. This curvature couldn’t be seen directly, unless you could somehow step out of three-dimensional space itself, but rather would make itself known through the behaviour of parallel lines and triangles. In a flat geometry (called Euclidean), if you draw a straight line and a point not on it, you can construct just one single line through that point parallel to the first line. For a saddle-shaped geometry, in contrast, there are an infinite number of parallel lines fanning out from that point, like the tracks out of a major city’s terminal train station. Moreover, while triangles in flat space have angles that add up to 180 degrees, in saddle-space the angles add up to less than 180 degrees. Yet another possibility, called positive curvature, resembles the spherical surface of an orange. Like the saddle-shape, its form could be seen only indirectly, through altered laws of geometry.

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When they talk about potential spherical or torus shapes is that an actual 3D shape or the surface area of that shape on a hypersphere (or hyper-torus etc)? How does string theory/M theory and the idea of 11 possible dimensions fit into this?



Is there any mathematial or physical breakthrough that would help decide which was the most likely?



My favourite would be the 1980s flat screen - disappear off one end and appear at the other....



thoughts? :)

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