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Comment by xyzzy4 | Hacker News Reader

How can a dimension be under a size? The four regular dimensions (3d and time) can go infinitely in any direction.

Well, how can you go on forever without falling off the edge of the Earth? The surface of the Earth two-dimensional, and approximately flat... but there's small positive curvature, so you're continuously wrapping around to a different part of the space. The surface of the Earth can be meaningfully understood as a two-dimensional surface that possesses finite dimension.

A three-dimensional space can be close to flat but have positive curvature as well (or negative curvature, for that matter). Some proposals give our universe positive curvature, rendering its space finite, though still stupidly-big. (I'm not aware of there being a final word on the subject, though). And if different dimensions can have different curvatures, some of them could be much smaller than others.

My understanding is that we have conducted experiments to measure the curvature of the universe, and come up with answers that are withing the experimental error of being flat. Without any theoretical reason why the universe should be flat, it is still possible that the universe is curved, but to slightly for us to detect, however, but the simplest interpretation is to say that the universe is flat.

Of course, if we a assume a multiverse with universes having regularly distributed, positive curvatures, then, the size of the universe would grow asymptotically as the curvature approached flat, so, statistically, life is more more likely to arise on the flatter universes.

Is it possible, or in any way related, that a small but non-zero curvature could be responsible for the accelerated expansion of the universe? I don't know, it's hard to picture. But I'm thinking the idea that you can look at a sphere as a 2 dimensional space that curves and eventually wraps around such that things going opposite directions on its surface eventually can run into each other again.

That would probably be more easily measured though. I don't know.

That's a remarkably close guess.

A small non-zero curvature would correspond to a non-zero cosmological constant, which is precisely what seems to be driving the accelerated expansion of the universe (there are other explanations but this is the simplest one).

I find these videos explaining things in an interesting way. https://www.youtube.com/watch?v=_k3_B9Eq7eM There's also a part 2.

Infinite is a big word :) . As explained above, the extra spacial dimensions would need to be compact, which implies them being finite in size. Perhaps the simplest example of compact manifold is the circle: you can imagine one extra dimension to be closed on itself. Going around the circular dimension one circumference length, you will find yourself where you started. The size of the dimension then refers to the characteristic size of the circle, i.e. the circumference.

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