= quote=

"If you take a cylinder and then fold it out you get a 'plane' (a flat rectangular piece). So drawing a triangle on the outside would give you a same triangle when folded back, measuring by the triangles interior angles. That's called a extrinsic type of curvature."

That is wrong. The surface of the cylinder has zero intrinsic curvature for just the reasons you state.

== End of quote

So when they write.

"Gaussian curvature is however in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature.

On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature."

You read this is a expression describing a two dimensional surface, right? It is a tricky subject, and one I haven't used myself.

(As for 'Euclidean geometry' I was referring to the type we used, and still use in school, before we get into SpaceTime and modern physics, topology, etc, suspecting that was how Euclid thought of it too?)

==

You know, rereading it it becomes even weirder. If we take the cylinder and assume it to describe some universe it seems to state that you can have a universe shaped as a cylinder and find it to be the same shape as when 'flattened out' measuring intrinsically. And then you say, if I got you right? That this only can be true in two dimensions. So what would I find, practically, measuring the same in three?

It also assumes 'ideal surfaces' it seems? How would I be able to be to measure anything, if I was part of that 'ideal surface'? To assume a 'ant' is either to introduce a third dimension, or mixing this subject with scales. But it's not about scales at all, well, as I read it? To me it's pure mathematical concepts?

Hopefully you will have a analogue, without the mathematics, that makes sense. Because practically, and as I think, we're part of the dimensions we measure, so when we find 'space' to 'bend' we should be 'bending' too, as a integral part of those 'dimensions'?