system
Given the following post, respond as if you're commenting on Reddit:
human
what does it mean for the universe to have a shape?

I’ve read different articles discussing theories of the shape of the universe but I can’t wrap my mind around all of the jargon explaining what that means and why it’s important to study. help!
gpt
This is a question that arises out of mathematical topology and geometry, but has relevance to astronomy, cosmology, and our understanding of how the universe works.

Imagine a group of 2-d people who live on a 2-d surface. There are certain properties about different surfaces that have meaningful consequences for those people. 

For example, suppose the 2-d people lived on the surface of a sphere. If a 2-d person starts walking along the equator and never changes direction, they will eventually come back to exactly where they started. This cannot occur if the 2-d people live on a different type of surface. If they lived on a flat, infinite sheet (such as a Cartesian plane), then they would start walking in any direction and they would never come back to where they started. There are other models too- such as a torus (a doughnut shape). If our 2-d people lived on the surface of a doughnut, then they could walk in a straight line and come back to where they started, and it would even be possible for two people to walk in a straight line and never cross paths- something that you can't do on a sphere. 

This is actually already relevant to astronomy. It is known that some stars and galaxies appear at two different places in the night sky. One plausible explanation is that we live in a "sphere," and that the light from such a galaxy had circled the universe so we see multiple copies of the same galaxy here at Earth. (The real explanation is gravitational lensing, however.)

Suppose our 2-d people live long enough to develop mathematics and topology, and they understand the difference between living on a flat plane, a ball, or a torus. Now they want to know what type of shape they live on. Well- it's not easy for them. We have the benefit of an "extrinsic" view of their world and we can consider "extrinsic" properties- in particular, we are 3-d people and can see how their 2-d surface is embedded in 3-d, so we can easily just look say that this is a ball versus a flat surface. Instead, the 2-d people have to develop some measure of "intrinsic" properties of such surfaces. For example, a really smart 2-d person named Riemann might realize that if you live on the surface of a ball, it's possible to draw triangles whose angles sum to greater than 180 degrees. Imagine you start walking at the equator and go 1/4th the distance around the ball, then make a 90 degree turn left and walk another 1/4th distance to the north pole, then make another 90 degree turn left and walk back to where you started. You have just traced out a triangle on the surface of the ball whose internal angles are each 90 degrees. This is impossible on a flat surface, so the 2-d people can actually cleverly measure and start to rule out what kind of surface they live on.

Now you see where this is going! Imagine that we are 3-d people (we are) who can be thought of as living on a 3-d surface that is embedded in some larger 4-d space. It is hard for us to visualize, because we are not used to thinking of ourselves as living on a 3-d surface, but that is the equivalent situation. Just like the 2-d people, we can come up with intrinsic measures of different types of 3-d surfaces, and start thinking about how to rule out different kinds of 3-d surfaces that we might inhabit.

If you are interested in reading more, a great introduction to the topic is *The Shape of Space* by Jeffrey Weeks. You can also see that author give talks on Youtube if you search.