Question

If you draw a big triangle in Earth 2D surface you will have an approximated spherical triangle, this will be a non euclidean geometry.but from a 3D perspective, for example the same triangle from space, it could be depicted as euclidean in 3D.Then why we talk about non-Euclidean instead of adding 1 dimension?It is easier the non-Euclidean approach? I don't see how!Sorry if the question is perhaps naive, but this is a long time doubt for me, so I will be grateful for an answer.

Answer 1

Non-Euclidean geometry is a mathematical framework that allows us to study curved spaces without the need for an extra dimension.

Step-by-step explanation:

The reason we talk about non-Euclidean geometry instead of adding an extra dimension is because the concept of non-Euclidean geometry allows us to describe and understand curved spaces without the need for an additional dimension. Non-Euclidean geometry is a mathematical framework that breaks away from the assumptions of Euclidean geometry, where the rules of straight lines and angles summing to 180 degrees no longer hold.

When we look at a triangle drawn on the surface of a sphere, it appears to be curved in 2D, which gives the illusion of non-Euclidean geometry. However, if we were to view the same triangle from a 3D perspective in space, it would appear to be Euclidean. The difference arises from the fact that we are looking at the triangle from different viewpoints, one on the curved surface of the Earth and the other in the 3D space.