The three main models of the universe are based on curvature: zero curvature, positive curvature and negative curvature.
A zero curvature would mean that the universe is a flat or Euclidean universe (Euclidean geometry deals with non-curved surfaces). Imagine space as a two dimensional structure -- a Euclidian universe would look like a flat plane. Parallel lines are only possible on a flat plane. In a flat universe, there is just enough matter so that the universe expands indefinitely without reversing into a collapse, though the rate of expansion decreases over time.
If the universe has a positive curvature, it’s a closed universe. A two-dimensional model of such a universe would look like a sphere. It’s impossible to have parallel geodesics (straight lines on a curved surface) -- the two lines will cross at some point. In a closed universe, there is enough matter to reverse expansion. Eventually, such a universe will collapse on itself. A closed universe is a finite universe -- it will only expand to a certain size before collapsing.
Negative curvature is a little trickier to visualize. The most common description is a saddle. In a negative curvature model, two lines that would be parallel on a flat plane will extend away from each other. Cosmologists call negative curvature models of the universe open universes. In these universes, there’s not enough matter to reverse or slow expansion, and so the universe continues to expand indefinitely.
Does this mean space is shaped like a flat plane, a sphere or a saddle? Not necessarily. Remember that space-time is measured in four dimensions, which reduces the usefulness of two-dimensional examples. And there are many competing theories about what the ultimate shape of the universe actually is.
One possible shape is the triple torus. At first glance, the triple torus appears to be an ordinary cube. But each face of the cube is glued to the face on the opposite side. Imagine that you’re in a spaceship that’s flying inside a large cube. You head toward the top of the cube. You wouldn’t smash yourself flat once you made contact. Instead, you’d appear in a corresponding spot at the base of the cube. In other words, you’ve gone up through the top and came back in through the bottom. If you traveled far enough in any direction, you’d eventually come back to where you started. This isn’t that foreign of a concept, since on Earth if you travel in a straight line, you’ll eventually come back to your starting point. You’ll just be very tired.
Another shape is the Poincaré dodecahedral spherical shape. A dodecahedron is a 12-sided object. The Poincaré variation has surfaces that curve outward slightly. What’s puzzling is that the projected size of this universe is smaller than the area we can actually observe. In other words, our visibility exceeds the boundaries of the universe. No problem, say the cosmologists. When you look at a distant galaxy that would seem to lie beyond the boundaries of space, you’re actually experiencing the wrap around effect described above. The galaxy in question would really be behind you, but you’re looking through one face of the dodecahedron as if it were a window. If you could see far enough, you’d be looking at the back of your own head.
Dizzy yet? There are many other theoretical shapes the universe could take, but most don’t fit the evidence we have so far. What is that evidence, and how do we gather it? Find out in the next section.