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For hypothetical beings living in a the surface of a sphere, it is possible to determine if their world is Euclidean or curved and closed by making triangles and measuring the sum of the angles (alternative measure the circumference of circles). For a large sphere they could maybe not notice it at first, but by making the triangle larger and larger they would eventually determine that they are not in an Euclidian space (without the necessity of going to the end of the universe to check).

The same idea of drawing 2d shapes to determine the shape of the universe to works in higher dimensions.

However for beings embedded in a 1+1D line it is not totally clear if there is a way to know if the universe loops back on itself.

Is there a way to determine if the universe is finite in a 1D world without the need to walk or send a signal that loops-back on the whole universe?

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    $\begingroup$ I don't think you can, but I might be wrong math.stackexchange.com/q/99365 $\endgroup$
    – user338734
    Commented Jun 23, 2022 at 21:20
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    $\begingroup$ There is no spatial curvature in 1-dimenaional space, but there can be spacetime curvature in (1+1)=dimensional spacetime. $\endgroup$
    – Buzz
    Commented Jun 24, 2022 at 2:05
  • $\begingroup$ @Buzz if there is only space-time curvature, could they notice the shape of the universe? $\endgroup$
    – Mauricio
    Commented Jul 4, 2022 at 11:14

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There is no intrinsic curvature in 1D. If you work out the math you find that there are $n^2(n^2−1)/12$ independent components of the curvature tensor in $n$ dimensions. For 1D that is zero, and for 2D it is just 1 component.

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Are you assuming that the 1D beings can only learn by sensory stimulus? Assuming they have sufficient mathematics, why couldn't a 1D being determine that it was living in a single dimension of a larger universe? We have developed many theories that consider many more than 4 dimensions, even though we can only sense 3 (or 4 if you include time as something that can be "sensed" through stimulus).

So don't quickly dismiss the ability of your 1D beings to be mathematics wizards who can work out complex motion in 3 dimensions just like we have come up with stuff we can't really measure directly. For example, torque relies upon the right hand rule (in cartesian coordinates), but does not depend upon some physical quantity that is in a direction perpendicular to the rotational motion. This is a convention of mathematics (well, sort of...)

But the answer to this question has HUGE implications about the nature of the universe, the existence of God, and the station of humans in the grand scheme of things. It is more than just an simple epistomological question about mathematics in R1 or some other R.

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