Question

You can “unfold” a cube to make six squares, shaped like a cross. You can unfold a cylinder to make a rectangle and two circles. Is it possible to unfold a sphere? What would an unfolded sphere look like?

Answer 1

As John Steele mentioned, the reason that we can’t ‘unfold’ the surface of a sphere (that is, as I take your meaning, place it onto a flat surface) has to do with the Gaussian curvature of the surface. Bending or even tearing the surface into pieces won’t change this curvature. The surface of a sphere and that of a flat surface have fundamentally different Gaussian curvatures which cause this to be impossible.

Consider extending some radius outwards from a point and drawing a circle at this radius. On a flat surface (ie euclidean) we measure this to be of length 2πr. On the surface of a sphere however we will always measure a smaller length than this. As an extreme example to illustrate this is true we can imagine starting at the north pole and having a radius that extends all the way down to the south pole - at this fixed radius the circle would have zero length - quite different than the same radius in euclidean space.

So if we want to press some section of a spherical surface onto a flat surface it becomes apparent that we will need to tear it at some point because we have a smaller circumference on our section than the flat surface has. What Gaussian curvature gets at however is that this phenomena exists in the space itself - down to an infinitesimal limit. We wouldn’t just have to tear our surface along one point to make it flatten - it ends up that every point in that space would have to be torn. I’m sure that you could imagine how this is problematic to our ideal ‘unfolding’ of the surface - its not much of a transformation if we need to tear it up into infinitesimal pieces.

https://www.quora.com/Why-cant-you-unfold-a-sphere-Isnt-the-surface-of-sphere-two-dimensional