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Scenario: A person lives in a 1-dimensional universe that comes to a 90 degree turn. Everything follows the line, and still only exists with one dimension at a time (a left to right line has no front/back nor up/down).

Question: Is this a 1-dimensional universe, or a 2-dimensional one?

It makes the most sense for this to be 2D, as the universe exists on an $xy$ plane. On the other hand, nothing has area, only one direction or the other.

Community
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Tyler Harper
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    If the 90-degree turn is observable to an inhabitant, it's not a one-dimensional universe. The turn only exists in the higher-dimensional space in which the one-dimensional universe is embedded. – chepner Aug 04 '23 at 13:04
  • If you want to have some in-1D-universe effect of the bend while still being strictly 1D, (or 1+1D incorporating time), you could have a gravitational or stress-energy peak in the neighborhood of the bend. Momentum through the "bend" might not be conserved, or there might be some compression / stretching / stress on 1D "objects" traversing it, but 1D observers couldn't measure the angle or direction of the bend, merely the intensity & extent of the stress. Compare https://physics.stackexchange.com/questions/566948/why-does-gtr-not-need-a-higher-dimension-to-describe-the-bending-of-spacetime – Sarah Messer Aug 04 '23 at 21:51

1 Answers1

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It is a 1D manifold. It only takes one coordinate to smoothly label every point.

Furthermore, all 1D manifolds are intrinsically flat. They can have extrinsic curvature, as you described here, but there is no measurement purely inside a 1D manifold that can detect the extrinsic curvature.

Dale
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    I was gonna say something like this. The turn wont exist in a 1d world without adding a dimension. But if the 1d world has a footprint on a 2d world it might matter to the latter. Like an electric cord, it can be straight, take corners or get tangled and it doesnt matter internally, but it matters to those tripping over it. – Dor1000 Aug 04 '23 at 12:03
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    Notably, this answer invokes topology, which is the mathematical study invented to be able to answer questions like these meaningfully. While "dimension" could have many meanings, including ones that lead you to find the answer to your question is "2 dimensions," the topological concept of dimensionality is quite robust so I'd say most mathematicians and scientists would likely use it, as Dale did. It would also be robust in the case where we say there is some additional connectivity going on in the bend (in which the answer may change) – Cort Ammon Aug 04 '23 at 16:03