[roywiggins]

It's like taking a bed apart and rearranging it into two beds, each identical to the original bed, without adding any more material.

[paulpauper]

Oh that is pretty cool. I wonder if it could be proved with differential geometry for the sphere, which has a simple paramertization

[Smaug123]

The ability to do it is equivalent to the axiom of choice, so my guess is going to be "not without considerable effort".