TIL that Gavin Theobald has produced several dissections from a triangle to a square on the surface of a sphere. Which would make pretty cool 3D-printed desk gizmos: make a ball with a triangular sector removed from one pole, a square sector removed from the other pole, and a set of wedge-shaped pieces that can be used to fill up either hole.
http://www.gavin-theobald.uk/HTML/Spherical/T90-S112.5.pdf
http://www.gavin-theobald.uk/HTML/Spherical/T100-S120.pdf