# Can 1D beings in 1+1D determine if they are in a curved universe?

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?

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

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.