Is there a standard definition of a given polyhedron using only intrinsic properties (those which can be measured by a 2d being living on its surface) and particularly angular deficit at a vertex (plus definition of relative position of vertices)?
My motivation: In physics, general relativity predicts that a toy 2+1 spacetime is flat everywhere except for the point masses which create angular deficit proportional to their mass. Flatland with one point mass is like surface of cone. When one adds other point masses the Flatland can be folded to a (convex) polyhedron (then there is the constraint on total masses, since total angular deficit is 720 degrees). In a sense, I try to imagine what geometry Flatlanders on a polyhedron could create.