I was looking at this article at mathworld where it has mentioned -
"This coordinate chart is not valid on the whole globe, since it doesn't give unique coordinates at the north or south pole (which way is east from the north pole?)."
I could not understand this logic/ any motivation ?
What error will we make if we consider the whole globe as the coordinate chart?.
Also from Wiki -
"As for the circle, one may define one chart that covers the whole sphere excluding one point. Thus two charts are sufficient, but the sphere cannot be covered by a single chart."
This example is historically significant, as it has motivated the terminology; it became apparent that the whole surface of the Earth cannot have a plane representation consisting of a single map (also called "chart", see nautical chart), and therefore one needs atlases for covering the whole Earth surface.
Any Mathematical visualization to this or a counter example would be helpful?.