Why is space three-dimensional, and not two-dimensional, or four- or five- or more-dimensional? I mean the extended space dimensions, not the micro dimensions. Why does it have a certain number of dimensions at all?
This is a plausibility argument, it doesn’t tell us why three, but why it might have a certain number of dimensions vs an infinite number of dimensions, or a variable number of dimensions depending on local conditions.
Imagine the particles of space are point-like (space could continuous – but for the sake of argument here), and they attract and repel each other so that a few nearby makes for attraction but alot nearby and they start repelling. Like the classic Game of Life >, except instead of multiplying vs dying, they’re attracting vs repelling. Or like people who prefer a small group to a large party – imagine how much harder that would be if the room weren’t two dimensional but three- or four- dimensional.
So the particles of space could line up single-file in a row, each mutually attracting its two nearest neighbors. Then space would be 1-dimensional. Imagine small metal balls. But you jostle them a bit and they attract and coalesce into a two-dimensional arrangement, like people who prefer more neighbors at a party. In our scenario, there’s more attraction cuz more nearest neighbors, but also a little repulsion cuz also more nearest neighbors. Jostle them some more and they coalesce into a more compact three-dimensional clump; at this stage there’s starting to be more repulsion cuz so many are near by. Now we jostle them more to try to get an even more compact four-dimensional clump of balls. But imagine at this point we’ve crossed a threshold – each one has enough nearest neighbors now that the repulsion would exceed the attraction, and they fight it and stay three-dimensional.
This is a plausibility argument for why space could be a certain number of dimensions, no more or no less. And what it means for space to have dimensions. The particles of space we model as tiny multi-dimensional spheres, and they self-assemble into something with a number of dimensions determined by mutual attraction and repulsion.
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