Question: Do we live in a Curved Space? What is the shape of the universe? What do you expect to find outside the universe?

David Freeborn answered on 24 Jun 2013:

Hi Sanban,

This is a good question, with an answer that might surprise you.

The most accurate answer is to say that the Universe is shaped like a doughnut, (we call this shape a “torus”), a ring with a hole in the middle, like this:

That might sound odd, but there’s some pretty good evidence that it’s true.

Roughly speaking, cosmologists talk about three general different types of geometry the Universe might have, which they wrongly call “flat”, “spherical” or “hyperbolic”. The three shapes correspond to different amounts of matter and dark energy in the Universe, which effects the geometry in different ways. I’ll explain what cosmologists mean by these terms (the names aren’t quite what you’d think):

(1) A flat Universe (the bottom of the three pictures) means that the geometry of the Universe behaves in the way you’re used to, for example the angles in a triangle add up to 180 degrees.

(2) A hyperbolic Universe (middle picture) means that space curves outwards. That means that the angles in a triangle add up to more than 180 degrees, as you might be able to see in the picture.

(3) A spherical Universe (top picture) means space curves in on itself. This means that the angles in a triangle add up to less than 180 degrees. It doesn’t necessarily mean that the Universe is a perfect sphere though, so this is a confusing name.

Now, we have a lot satellite data from two satellites called COBE (Cosmic Background Explorer) and WMAP (Wilkinson Microwave Anisotropy Probe). By measuring distances between objects, they’ve been able to find out what the geometry of the Universe is. The result surprised a lot of physicists: we found that the Universe is almost perfectly flat: to within 0.4%.

So, how can the Universe be both flat and doughnut shaped?

Well- there are certain types of shape that correspond to others. This comes from a branch of mathematics called topology.

For example, you can never turn a flat piece of paper into a sphere (or vice versa- as you’ll know if you’ve ever tried to flatten an orange peel). Nor can you turn it into the “hyperbolic” shape above. If you try to bend it down the middle, you’ll find you can’t bend it the other way.
So spheres, “hyperbolic planes” and flat sheets are all “topologically distinct”.

But you can turn a flat piece of paper into a torus- at least if it’s flexible enough. Curl the paper round to make a cylinder, and then bend the cylinder round back onto itself (this is easier with a rubber sheet than with paper!). So a Torus and a sheet of paper are topologically the same, and in fact, angles on triangle on a Torus do add up to 180 degrees just like on a flat piece of paper.

We know that the Universe’s geometry can’t be “flat” like a sheet of paper, because space doesn’t have any edges. If you keep travelling far enough, you end up back where you started.

How do you do that with a flat piece of paper? You bend it round so that every edge touches another edge. The way to do this makes a torus.

So, amazingly enough, we know that the Universe has a “doughnut” type torus shape!