Take out your piece of paper with two parallel lines on it. Go ahead, dig it out of the trash. Wrap one end around to meet the other, making a cylinder. Carefully observe the parallel lines — they remain parallel, don’t they? That’s because cylinders are flat.
You heard it here first: Cylinders are flat.
There’s an important distinction between geometry, the behavior of parallel lines, and topology, the way a space can get all twisted up. While the geometry of the universe is very well measured (again, it’s flat), the topology is not. And here’s a bonus fact: not only can we not determine the topology of the universe from observations, but there are also no laws of physics that predict or restrict the topology.
With your 2D piece of paper, you can connect the ends a few different ways. Connect one of the dimensions normally and you have a cylinder. Flip one edge over before connecting and you’ve made a Mobius strip. Connect two dimensions, the top to the bottom and one side to the other, and you have a torus (aka a donut).
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