When cosmologists talk about the shape of the universe, what does that mean?
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You are sitting on the surface of a sphere (or what we can treat as a sphere, for the sake of argument).
If you draw a small triangle, it will come very close to obeying the laws of Euclidean geometry (the geometry you probably learned in middle or high school). Its angles will sum to 180 degrees, it will have area (1/2)*base*height, and so on. In other words, even though you’re on a sphere, your *local* space *looks like* a plane, in the sense that it approximates a plane better and better the more you zoom in.
But if you zoom out, geometry gets weird as the curvature of the Earth starts to become relevant. In very large triangles on the Earth, the angles *don’t* sum to 180 degrees, the area *isn’t* (1/2)*base*height, and so on. On a sphere, all sorts of things in Euclidean geometry break down. There are, for example, no such things as parallel straight lines on a sphere! All “straight lines” are great circles (like the lines of longitude, but not those of latitude) that eventually meet back up somewhere else on the Earth.
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The Universe as a whole behaves similarly, just with some extra coordinates. Rather than a 2d sphere that approximates to a 2D plane, it’s a (???) that approximates to flat 4D space-time if you zoom in enough. What’s at stake here is what happens when you *don’t* zoom in.
For example, if you fell into a black hole, *your local space would still look like normal 4D space-time* to you. But the space everywhere *else* would look terribly twisted. Similarly, from outside a black hole, your local space still looks like normal 4D space-time, but the space near the black hole looks very twisted. This is, in essence, the same thing as how you and someone on the other side of the Earth can both look “up” but be looking in totally opposite directions.
The “shape” of the Universe is the question of what it looks like if you zoom **out**. And we don’t know the answer to that question. We know it looks flat out to some pretty large distances (except for the local bending around massive objects), but we don’t know whether that continues forever. We don’t even know what “kind” of shape the Universe is. Is it like the surface of a sphere? Like a flat plane? Like a stranger shape like the surface of a torus (donut shape)? We don’t know.
(These two questions – how the geometry works and what “kind” of shape it is – are related but distinct, and in the full mathematical treatment we’re talking about geometry and topology respectively. But the difference between those is hard to ELI5.)
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