The Shape of the Universe (is there a boundary to the universe?)
You may have wondered at one point, is there an edge to the universe?
On the one hand, the universe might simply be infinite in space, simply extending forever. Such a scenario is a perfectly good solution of Einstein’s equations of general relativity, and is consistent with all observations of the universe. There may be a legitimate philosophical objection to the existence of an actually infinite physical quantity, but I won’t get into that here.
Broadly speaking, an infinite universe is called open, and a finite universe compact or closed. And our universe could just as well be closed as open. A sphere, for example, is closed: you can go from every point to every other point, and there is no edge.
In fact, it’s possible that the shape of the universe is spherical – but a three-dimensional rather than two-dimensional sphere. Just as one can sail around the earth, it may be possible (in principle) for a spaceship to fly off in one direction and return to the same place coming from the other direction. This may seem bizarre, but it’s only confusing because our brains aren’t wired to visualize three-dimensional spheres as they can two-dimensional spheres. We wouldn’t necessarily notice the spherical shape of the universe, just as a person standing on earth cannot notice that they are standing on a sphere.
How would we know if the universe was shaped like a sphere? We might be able to measure its curvature. Think about the example of the earth again. Suppose I was standing on the South Pole, and wanted to test the hypothesis that the earth was flat. And suppose I knew of two very distant lighthouses – I knew their distance from each other, and from the South Pole. If the earth was flat, I could use simple geometry to find the angle I would expect to measure between the incoming light beams. But a sphere, that angle would be larger. (See Wayne Hu’s tutorial for an image.) This is because if you connect three points on a curved surface, the sum of the three angles formed is no longer 180 degrees.
So, by measuring the angles formed between incoming light signals from known sources, I could determine whether or not the earth was flat. The universe works in a similar way, except with one more dimension of space. The cosmic microwave background, background radiation filling space, gives us a temperature map of the early universe at a distance of almost 13.8 billion light-years (CMB light has been travelling for nearly the age of the universe). The physics of the CMB is well-understood, and we know the actual physical size of “hot spots” and “cold spots” in the CMB, so by measuring how large of an angle they take up on the sky, we can determine the curvature of the universe. As far as we can tell, it’s a flat universe. But this may only be because we lack the precision to distinguish curvature, just as a terrestrial observer would need precise instruments to discover the globe beneath them.
Now, if we did measure this kind of curvature in space, it would not necessarily mean that the universe was spherical. It could have a more complicated shape, in which our local region appears to be curved like a sphere, just as a nearly flat sheet with many tiny ripples could look curved when viewed in a very small region. In fact, this is exactly what the universe is like. Small hills and valleys in the gravitational field (which also lead to the temperature variation in the CMB) curve space locally, and a hill or valley stretching over the largest observable distances could look like a curved universe, and could obscure measurements of the global curvature just as mountainous terrain could screw up the earthly experiment described above. Fortunately, these variations in the gravitational field are tiny (one part in 100,000!) so only an equally miniscule global curvature could be hidden by them. In other words, a spherical universe would have to be enormously large to look so nearly flat from our vantage point.
We would have stronger evidence of a sphere-shaped universe if, for example, we observed the same galaxy cluster on opposite sides of the sky. In this case, we would be looking around the sphere on opposite sides, and seeing the same thing.
Note that in neither of these cases does the universe have an edge. General relativity – Einstein’s theory of gravity which describes the geometry of space and time, and has been confirmed with many experimental tests – does not describe space that way.
Kosmophilus 
Thanks Elliot for this thought-provoking exploration of a question I’ve had for a long time. (And I should say up front that I know very little about cosmology, so what I’m about to ask may be quite naive).
I’m still not sure in what sense a sphere lacks an “edge” in the sense relevant to the problem. Of course, you would never reach the “end” of a sphere and fall off, as you presumably would if the earth was flat – the place where the ships fall off the “edge.” But still, even if the universe is spherical, how can we explain its expansion, since our ordinary concept of something expanding requires that there be space around it for it to expand into. But if the universe is, physically speaking, all there is, then there could not be space outside of it for it to expand into.
In other words, if the universe is finite, then even if it does not have an “edge” it does have a boundary, no? But it’s hard to imagine how it could have a boundary since the only way we have of thinking of boundaries requires that there be something else limiting the object we’re thinking of. My problem in trying to think of this is compounded by the idea that the universe is expanding, because that seems to require all the more that there should be something outside the universe for it to expand into, which seems to be ex hypothesi impossible (the universe, after all, is supposed to be all there is, spatio-temporally, no?). Are you willing to say more about how to think about a finite and expanding universe without running into this problem?
I think the subtle part of the problem is that while the universe is expanding, but NOT in the sense of expanding into some external space in which it resides. So in what sense is it expanding? In the sense that the distance between any two points that are widely enough separated (eg. two galaxies) is increasing, even though those points are not moving. In other words, like the distance between two marked spots on a balloon is increasing when you blow it up. This statement doesn’t rely on the fact that the balloon is expanding into a larger 3D space. So if you were a two dimensional creature living on the balloon, you could measure the expansion by observing that points were moving away from each other. You wouldn’t need to know that the balloon was expanding in some larger 3D space. So this is the sense in which the universe is expanding. And just like the two dimensional balloon creature could not find an edge or boundary, we may not be able to find one for the universe.