I want to build a virtual world that feels like an unbounded flat plane but actually "connects back to itself" with the topology of a sphere. To do this, we can build the world out of polygonal sectors corresponding to the faces of a polyhedron (bordering each other in the same configuration). This works nicely except that each vertex of the polyhedron becomes a weird point, around which a small loop has a total turning angle less than 360°. (For example, if the polyhedron is a cube, each vertex is the meeting point of three square sectors, so you can walk all the way around it while making just three quarter-turns.) We can still render these locations using a portal technique and put something like a tall tree at each one to obscure the visual discontinuity.
To make these "corners" less conspicuous, I'll use a polyhedron with a large number of vertices. But if we didn't care about that, how few corners could we theoretically get away with? I think the smallest polyhedron that makes sense with my construction is a double-sided triangle, which has 3 corners. But what about a space like a cylinder with each end pinched together (2 corners), or the one-point compactification of a flat open disk (1 corner)? I can't visualize 3D embeddings of these that preserve flatness, but could there still be a corresponding "flat world" with just one or two singular points?
In order to answer this, I think I need to clarify the idea of a "manifold with corners", and that's where I'm having trouble. I've thought about cutting the corners off (leaving a small boundary loop) or rounding them off (leaving a small region with nonzero curvature) to obtain an actual manifold (on which the Gauss-Bonnet theorem might be helpful), but I'm not sure how to express the requirement that the resulting defect is still "point-sized", which seems important. Inside the flatworld, I want the singularity to appear as a "pole" standing on a point of the plane (as opposed to some larger object), and I think this is what could rule out the one- or two-corner possibilities.
I've convinced myself that 3 corners are necessary, based on a sort of unsatisfying description of what a corner is. I'd still appreciate an answer with more clarity on how to define a topological space that's "flat except at corners" and what's wrong with the proposed cylinder/disk quotient spaces.
The idea is: For a corner to be "point-like", you should be able to walk a small counterclockwise loop around it, keeping it on your left and ending up with a total left turn angle of $\theta>0$, which measures the angular size of the space surrounding the corner. Since $\theta>0$, the "deficit" $2\pi-\theta$ must be strictly less than $2\pi$. By the Gauss-Bonnet theorem, the deficit measures the curvature at the corner, and if the surface is flat except at the corners, the sum of the deficits must be $4\pi$. (We can verify this for various polygons.) But this means we need strictly more than 2 corners.