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It’s kind of weird thinking about the universe having a shape. It’s the universe, right?

It’s … universe … shaped ... But it does have one. We just aren’t sure what that shape is.

And some cosmologists have proposed that of all things, it might be shaped kind of like a donut. When we’re talking about the shape of the universe, we’re really talking about its topology. The topology of an object has nothing to do with how many sides it has, or where parts of it are relative to one another.

It only depends on the number of holes: a sphere has none, a donut has one, and so on. A sphere and a donut are both round, but they have different topologies because there’s no way to take a sphere and stretch or smash it to make it look like a donut without, shall we say, “tearing it a new one.” On the other hand, a lot of objects we think of as clearly being different shapes have exactly the same topology. A coffee mug has one hole in the handle, so topologically speaking, there’s no difference between a mug and a donut.

There are a lot of proposed topologies that the universe could possibly have, from plain shapes with no holes to some that are much more complex. But one of the most common ideas is what’s called a 3-torus. It’s impossible for our puny human minds to visualize what that looks like from the outside, because you’d need a fourth spatial dimension.

A line is one dimensional. A flat plane: two. And with the third dimension you get depth.

Usually, when we talk about the fourth dimension, we mean time. But in this case, it’s a fourth spatial dimension — the next level up from 3D. And as beings who’ve lived our whole lives in 3D space, we can’t really imagine what 4D would look like.

That doesn’t mean you can’t model a 3-torus, though! You just have to step everything down a dimension. You can take our universe, with its 3 spatial dimensions, and model it as a two-dimensional flat plane — think of it like a piece of paper, except pretend it stretches to infinity along both axes.

If you take that piece of paper and wrap it into a cylinder, it stays infinite in one direction but becomes finite in the other, because eventually you loop around the cylinder. If you then wrap the ends of the cylinder together, making it connect along both axes, you have yourself a 3-dimensional version of a 3-torus. There you go.

Donut universe. Topology is different from geometry. It doesn’t care if or how space is curved, or about the distance or angles between different objects.

Geometry does; it knows the difference between our mug and donut, and we’re actually pretty sure we know what the geometry of the universe is. Measurements have shown it’s almost definitely flat, as opposed to positively or negatively curved. Which, again, gets a little brain-break-y, because “flat” is not normally an adjective we use to describe things in 3D.

But cosmologists talk about 3D space as being “flat” or “curved” because it’s the best analogy we have for what we’re trying to describe, which basically boils down to this:. If you shine two laser beams in parallel, do they stay parallel? Or do they eventually cross or diverge?

When we say the universe is flat, we mean that the beams stay parallel — space isn’t warped in a way that would curve them toward or away from each other. The ‘donut’ universe is actually a topology with a flat geometry, which our simplified 3D model can't, well, model. But the idea is that instead of a simple plane that stretches to infinity in 3 dimensions, it connects at multiple points.

In other words, it’s finite. But it mimics an infinite universe because light wraps all the way around the shape, so you get the effect of standing in one of those infinity light boxes. The 2D version of this is a classic video game where, if you move to one side of the screen, you pop up on the opposite side with the same speed and trajectory.

In 3D space, if the universe were incredibly small, and you looked directly up, you’d see the bottom of your own feet. Or if you looked over your shoulder, you’d see yourself repeated in an infinitely long line. But our universe isn’t that small.

So how could we possibly tell if we’re living in a 3-torus? Basically, astronomers hunt for recurring patterns in astronomical images -- the same cosmological structures showing up in multiple parts of the sky. That would mean light is wrapping around the universe, and we could run models on those patterns to determine its topology.

Or at least, we could if the true size of universe is smaller than the distance it takes the light of everything to reach us. Because remember, light takes time to get to us. There could be things out there so far away that even though the light coming from them has been traveling almost since the beginning of time, it still hasn’t reached us yet.

If there is stuff beyond where the earliest light we can see originated, the light wouldn’t have the opportunity to wrap around and create any repeat images. The universe could be a donut, and we’d never know. That said, results as recent as 2015, courtesy of the European Space Agency’s now-defunct.

Planck satellite, have failed to find any evidence of a 3-torus topology. Or any topology. So as far as we can tell, the universe is not a donut.

But what if it were? Well, in day-to-day life - even in certain fields of astronomy - it wouldn’t matter very much. Just like there’s a lot of physics we can still do with old-school Newtonian equations, changing how we think about the universe’s topology wouldn’t require rewriting the rules of stellar life cycles or planetary formation.

But there is one bit of cosmology a donut universe definitely threatens: inflation. Cosmologists think the universe suddenly expanded in size when it was super duper young. When we look at the light left over from the Big Bang — what’s known as the cosmic microwave background, or CMB — we can see evidence of that inflation.

But there’s nothing in the theory that would cause inflation to produce a universe big enough to house all the galaxies, yet small enough to see itself repeated within the observable horizon. So knowing the universe was a donut would force astronomers to find a new explanation for why the CMB looks the way it does. But one of the cooler consequences of a donut universe?

Somewhere out there, some of the oodles of galaxies would actually be the Milky Way itself. But we wouldn’t be able to tell just by looking at them, because they’d appear billions of years younger. So you could actually wave to yourself, but you’d be very dead before you received the message.

But hey, it’s the thought that counts. Thanks for watching this episode of SciShow Space. If you want to celebrate your love of the universe and drink coffee out of a topological donut every morning, we’ve got your covered!

Head over the DFTBA.com/SciShow to get your own topological donut! It’s a coffee mug. We hope you like it.

And thanks for supporting SciShow Space! [ ♪ Outro ].