This began as a response to conversation on Tumblr, in which someone claimed the universe was finite. A friend of mine responded:

theomenroom said:

Actually, if I understand space right (maybe one of my followers who studies physics can confirm?) the thing about ending up where you started if you go far enough is only true if the universe has elliptic curvature, and last I heard evidence suggested that it was flat overall (some parts are slightly elliptical, some parts are slightly hyperbolic, depending on the local balance of mass and dark energy or something and how that works out with general relativity, and overall it adds up to flat. of course literally all of that, except some of the geometry, is way over my head so)

jonesinforjosie replied:

We could have a positive curve universe, negative curve, or no curve. If I recall correctly, positive curve means it’s like a sphere ( if I’m wrong, positive means a saddle shape and negative means a sphere), and we are confident it’s not a sphere because if it were, we’d be able to see ourselves if we looked far enough in any direction?

theomenroom replied:

That’s true but the visualizations don’t make a whole lot of sense if you don’t know the properties they’re trying to describe. Positive curvature is elliptic and negative curvature is hyperbolic, yes.

Let’s say you and a friend are holding hands in a space with some curvature. There’s some distance between you, and you’re walking in the same direction at the same speed, exactly perpendicular to your arms, and each of you is following your own line. In a flat (Euclidean) space, which is the most intuitive, you remain the same distance apart no matter how far you walk.

In an elliptic space, no matter which direction you’re walking (as long as you’re walking the same direction), you’ll end up closer together just by walking forward, until you bump into eachother. If you and your friend are geometric points with zero width, you will eventually collide at the “pole” corresponding to whatever line your arms were on initially. Also, if you keep your arms on the shortest distance between you, the angle between your path and your arms will shrink as your arms end up leading you. Once you reach the pole, continuing onward and turning back around may be mathematically equivalent, depending on the construction of the elliptic space you’re in (though if it is, that’s because going in any direction and its opposite is always equivalent. believe it or not that makes the math simpler). If it isn’t, once you do you will again put space between you, reach a maximum distance at the furthest possible point from where you started, and then approach another pole (and then go back to where you started). You can sort of see how this will work by drawing great circles on the surface of a sphere (circles which share a center with the sphere; meridians are great circles, parallels are not), which is why this is said to be spherical

Hyperbolic curvature, on the other hand, does the opposite. If you and a friend walk like this, your arms will trail behind you, and will also have to get longer the more you walk; just walking distance adds space between the two of you (and the further you go, the faster it adds space). Like in euclidean geometry, and unlike in elliptical, in hyperbolic geometry this just continues without end. There is more and more space the further you go.

People call this “saddle-shaped” because drawing lines on an (idealized) saddle and then measuring distance along the surface of the saddle has some similar properties to this, but embedding something that infinitely expands inside a euclidean space that doesn’t is hard.

Carolyn (theomenroom)'s explanation was good, but I wanted to add some extra detail...

In general relativity, the “curvature of space” is determined by the mass density there. however, on enormous scales, we can safely model the universe as homogeneous and isotropic, which has the result that the curvature is basically the same everywhere.

When we talk about whether the universe is infinite or finite, we are talking about the topology of the universe rather than its curvature, which is a local property.

If you look up the shape of the universe on Wikipedia, you will find it awash with potentially unfamiliar mathematical terminology such as ‘metric space’, ‘compact’, ‘bounded’, etc. I’ll try and write it more simply here.

• The universe may be infinite, which is to say, there are points that are any distance apart, no matter how big.
• If it isn’t, it might have a boundary, or it might not. example of a 2D space that has a boundary: a disc. example of a 2D space that does not have a boundary: a sphere. Both of these have 3D equivalents.
• We think the universe does not have a boundary, because we have no idea what it could possibly look like, how anything would behave there, etc.
• So we’re left with a “compact metric space without boundary”. Add the requirement that it’s “differentiable” and we can call it a “closed manifold”.
• That, however, still leaves various possibilities - the universe may be “simply connected” like a sphere, or “multiply connected” like a torus.
• This is kind of whether there are any holes in the space in a certain sense. Basically if you draw any loop in space, can you shrink it down to a point? If you can, you’re in a simply connected space. if you can’t for some loops, you aren’t.
• The topology of a space doesn’t tell you much about the curvature in any particular place!
• For example, a regular polygon is topologically equivalent to a sphere. but the curvature of the space is 0 except at the points.
• However, constant curvature of a space does place limits on the kind of topology the space as a whole can have.
• The topology of the space gives it certain “topological invariants” and these can be related to the integral (basically like a sum) of the curvature at every point of the space
• Since we assumed the universe is homogeneous and isotropic, the curvature has to be the same everywhere. So whatever value we pick for the curvature will place limits on what topology the universe can have (e.g. sphere, torus, infinite)
• Please note: all of this is really hard to imagine because we, being creatures who perceive a 2D projection of a 3D world, are really not good at imagining a 3D ‘surface’ embedded in four dimensions!
• As noted in the previous post, if the curvature is positive, the universe has to be compact, whether or not it has a boundary. In fact it has to either be a 3-sphere or one of the “quotients” of the 3-sphere. This would mean that you can head out in one direction and go ‘around’ the universe and end up back where you started. I do not recommend trying this.
• If the universe is flat, it can be compact or infinite. If it’s infinite, you have Euclidean space. Otherwise, you can have things like 3D toruses or 3D Klein bottles. (you can think of the 3D torus as being a universe that repeats endlessly in every direction.) There are, apparently, ten different possible topologies for a flat finite 3D universe.
• If it’s negative, you’ve got hyperbolic geometry. Wikipedia says “There are a great variety of hyperbolic 3-manifolds, and their classification is not completely understood.” So that’s helpful. If you click on that link, it starts talking about knots and I stop being able to pretend I understand this.

so it’s pretty much as @theomenroom said but that’s a bit of extra detail.

Now, as for the global topology of the universe of a whole? We actually don’t know!

The problem is that the universe is really fucking big. According to current cosmological models, the parts of the universe we can see now comprise a big sphere 93 billion light years across. (Hold on, isn’t that bigger than the 13.8 billion year age of the universe? Yeah! Because space is expanding, and the light we’re receiving was emitted when the universe was a lot smaller. Also, relativity makes ‘now’ a kind of complicated thing. That distance is the comoving distance for the present time, if you’re worried.)

Now, if the entire universe was smaller than the observable universe, we might see stuff repeating in the sky, in increasingly early instances. But if the universe is bigger than the observable universe, we can’t necessarily tell whether it’s finite or infinite! Awkward.

What we do know is that the curvature of the universe is really damn close to flat. so we probably don’t have to worry too much about that spherical and hyperbolic stuff. But is it a 3-torus, or an infinite space, or what? We don’t know and maybe we never will!

Cosmologists are trying to work it out though. They try to work out what effect certain shapes of universe would have on the things we can observe, such as the cosmic micorwave background radiation. So far: inconclusive.

Another cool fact: in an infinitely big universe, there are, statistically, infinitely many copies of everything, though they’re arranged randomly. How far away is the nearest exact copy of the observable universe? We can estimate this! I’m trying to find a good source for the calculation, but an old scientific american article by cosmologist Max Tegmark says a copy of you within 10^(10^28)m, a 100 light year sphere (corresponding to a matching 100 years of history) within 10^(10^92)m, and the entire observable universe should have a copy within 10^(10^118)m. For comparison, the observable universe is a mere 10^27m across.

These numbers are hilariously big - if I wrote them out in this post as 10000… we’d need 10^28 bytes to write even the smallest distance (the one to an exact copy of you), which is to say 10,000,000,000 times the estimated amount of data on the entire Internet. I tried putting 10^(10^28) into Wolfram Alpha to convert it to light years, and the calculation returned no results.

If you got in a spaceship and tried to travel to the clone universe at a speed arbitrarily close to the speed of light, before you got even the unimaginably tiniest fraction of a percentage of the way, all the stars in the universe would burn out, the particles would decay (depending whether protons decay), and only black holes would remain; the black holes would decay into particles through Hawking radiation, and eventually the universe would contain nothing but photons that almost never encounter each other. You’re still not noticeably closer. By the time you get halfway to where the clone universe used to be, a new Big Bang might happen apparently (idk how they worked that one out). Meanwhile, the expansion of the universe is moving the clone universe even further away, not that there would be anything left of it by the time you get there.

An infinite space contains points separated by any distance, and it will contain infinitely many copies of our observable universe - as well as infinitely many near-exact copies that differ by only a few particles, and so forth.

Anyway, we don’t know the overall topology of the universe - it might not be infinite, or big enough to contain any such silliness as an exact copy of the universe.