Question

Show that the surface of a sphere can't be represented as a plane (?)

Answer 1

See explanation below

Explanation:

Sphere surface is a non ruled surface. That means that sphere is not "similar" to a plane. Locally, we can put a tangent plane in every point of sphere surface, that represents locally the spahere, but globally we cant do that. A cone is a different case. We can cut through circle of base and generatrix and we can to strech the cone in a plane. The same with a cylinder. But not with sphere. Based on "egregium theorem" of Gauss, we know that a surface is ruled if and only if his curvature is identically zero. All ruled surfaces are called developables

So we have to prove that curvature of sphere is not zero at any point on it. But we know that curvature of a sphere is `K=1/R^2` with `R` radius of sphere, so is different of zero and then Sphere is a not ruled surface.

For this reason we have to "approximate" sphere with other ruled surface, like cones, cilinders, etc in order to represent planes of Earth.

Hope this helps