What's up with the concept of "shape" applied to the universe? How can something with no practical boundaries have a shape?

So first, let's start off with a confession. No matter how many pretty pictures of spheres and whatnot that I put up, you're probably not going to really have a gut feeling about what it's really like living in a curved universe. If it makes you feel any better, I still can't really picture it.

The problem is that in order to explain the shape of the universe to three-dimensional creatures (including you, presumably) I'm going to need to use the old "rubber sheet analogy" of pretending you're a 2-dimensional ant walking around on a balloon or a rubber sheet. It's not ideal, but it's a good way to get your intuition warmed up.

Life as an ant

So you're an ant living on the sphere. This is a pretty good picture of how you live your life now, provided you keep your head perfectly level, and don't climb stairs or fly in airplanes. For those of you who are somehow missing the patently obvious, I'm talking about walking around on the surface of the earth.

As an ant, you don't look up and don't look down, just forward, backward, left and right, and you can only move in those directions. Also, for the purposes of this discussion, you get to walk on water, so that's pretty sweet. Your sole mission in your ant-life is to figure out whether your ant-world is flat or curved, big or small.

As humans, we know (at least, I'm assuming that the io9 audience is in more or less complete agreement) that the earth is round. Of course, we get to cheat. We can get into a rocketship and go in the "up" direction (a direction, I remind you, that doesn't exist for our 2-dimensional ant friends) fly all around the earth, and see that it's round directly.

Ants don't have this luxury, so how does he figure out that his world is round? If he has the time, he could walk and simply figure out how long he has to travel before he gets back to where he started. But this isn't going to be of much use for us humans to figure out whether our 3-dimensional universe is curved since it would require traversing the length of the universe. Supposing it's even possible (and it seems as though it isn't) it would take many billions of years to do it, and seriously, shouldn't there be a better way?

A more formidable problem is the fact that a Pac-Man universe — one in which you get back to where you started if you travel long enough — doesn't necessarily mean that the world is round.

Wha?

What shape really means

Let's stick around in ant-world for a while and figure out how Anty McAnterton really can figure out whether he's living on a round world or a flat one. It's all about geometry.

You learned when you were but wee tots all about how geometry is supposed to work, all thanks to Euclid: Triangles have 180 degrees in their interior angles. Parallel lines never meet, and so on. The thing is, Euclid isn't always right. Actually, the only time when Euclid gets it right is when space is flat.

Check out the diagram above. Draw a triangle on a flat plane and it makes 180 degrees, just as Euclid said it would. On the other hand, on a sphere, the angles make up more than 180 degrees, perhaps considerably so, depending on the triangle. It's like the old riddle about traveling 1 mile South, 1 mile East, and 1 mile North and ending up at the same point and subsequently being eaten by a polar bear. That sort of thing only becomes possible if you're on a sphere.

The smaller the ant and his range of motion is compared to the size of the ant-world, the less he notices this curvature. This, incidentally, is why we don't have any problems making a nice, flat map of a town, but it's impossible to make a perfect map of the entire earth. The curvature becomes really important. In other words, if the curvature of the universe is big enough, there's really no practical way of distinguishing between a flat one and a curved one.

But how would we, or the ants for that matter, figure out whether the universe is curved, even in principle? Using light. Suppose there were two super-civilizations very far from the earth — billions of light years — and very far from each other. If each of us measures the angle between the other two, and we send signals to each other so that we can add them up, we can figure out if they total 180 degrees, or more, or less.

By the way, you may be curious about the middle part of the diagram above, the one that looks a little bit like a saddle. It's just another way that the universe can be curved, but one that probably looks very unfamiliar. Just like a sphere gives you triangles with more than 180 degrees, a saddle (and frankly I can't even try to imagine a saddle in 3-d space) gives you triangles with less than 180 degrees.

How do we know?

Enough of antworld. Why should our universe be curved at all? For the same reason that any sort of space-time gets curved in our crazy universe: there's stuff in it. In case you've forgotten, one of the major predictions of Einstein's theory of general relativity is that mass and energy curve space and time.

The same is true for the universe — our human 3-d universe — as a whole. If there's too little stuff the universe is saddle-shaped. Put in too much, and the universe is spherical. But put in just the right amount (and this seems to be the case for us) and the universe will be flat. As we used to rather cheesily put it, "Density is destiny."

One reason that cosmologists actually care about whether the universe is flat or curved is that we can do the problem the other way around. If we know that the universe is flat, for example, then we know the exact density. As it happens, there's an astonishingly good way to measure the curvature of the universe, and it has nothing to do with super-civilizations billions of light years away.

There's a ready-made way to do geometry on the scale of the entire universe. For about 380,000 years after the big bang, the universe was an insanely hot mess of ions, electrons, and photons, all coupled together like a fluid. It sloshed back and forth, just waves do when you play in the tub (though you've perhaps been too distracted to notice). After 380,000 years, the universe got cool enough that the atoms could become neutral and since photons don't care much about neutral atoms, what we now see in our Cosmic Microwave Background is really just a relic of those early days. Some spots are slightly hotter than others (by about 1 part in a hundred thousand) and some spots are slightly cooler.

However... how we see those early days depends strongly on the shape of the universe. Check out the diagram above. A spherical ("closed") universe makes all of the bumps and wiggles (really, the hot and cold spots) look larger, and a saddled ("open") universe makes all of the bumps and wiggles appear smaller. As it happens, every measurement we've yet made puts as exactly flat, at least to within the very small error-bars. Good thing, too, since that's also what theory predicts.

That, by the way, is one of the big reasons that we're so sure that so much of the universe is "missing" in the form of dark matter and dark energy. And I know what you're thinking: "Really? You're fitting the size of a few blobs and expect us to believe in dark matter and energy?" Well, hypothetical (but, if history is any indication, far too real) io9 reader, I do expect you to believe it. The reason is that we're not just fitting a few blobs, but we're fitting the spectrum of the background over the entire sky, and it produces an amazing fit.

Pac-Man lives in a flat universe, too

But even if the universe really is flat, that doesn't mean that it has to be literally infinite. We could still live in a Pac-Man universe. You can think about it from our ant friend's perspective. Suppose he's living on a flat piece of paper that I have conveniently rolled into a tube. There's no way that he can tell by anything I've described that his universe is anything other than flat. Draw a triangle on a piece of paper, roll it up, and then whip out your protractor. Still 180 degrees. However, if the ant walked for long enough, he would still come back to where he started.

General relativity says nothing about whether the universe folds back on itself. If you're looking to impress with fancy talk, you could say that this is because relativity tells us about the "geometry" of the universe, not the "topology."

There's an awesome consequence of all of this. In principle, could imagine looking in two different directions of the sky, and because the universe loops back, we'd see two images of the same galaxy. In the real universe, we don't see anything so dramatic, but that hasn't stopped people from looking for "circles on the sky." In 1998, physicist Neil Cornish, then at Cambridge, and his colleagues proposed searching for repeating patterns in the cosmic microwave background. So far, nothing.

In other words, if the universe isn't infinite, it's pretty damn big.