EDIT: My guess to the below question is that no you can't triangulate a triangle because a legitimate triangulation each edge can only be linked up to exactly two distinct faces, so if you just have one triangle, each edge would be linked up to one face (the face of the triangle) I'm really getting confused when it comes to triangulation. I have realized now why you can't triangulate a torus with just two triangles (I had thought before you could just take the regular square and split it down the middle in to two triangles - but I see now these triangles would not be coherently oriented which would mean it wouldn't be orientable but the torus is orientable, so you need a lot more triangles) but I am wondering - if you just have a single triangle, does that count as a triangulation? Like say a triangle, regular 3 sided triangle, say arrows pointing all in the same direction around the triangle giving it an orientation - does that mean it's all ready triangulated? Everything I see talks about having to have even sides to be a surface and to be triangulted, etc. But I don't really see why and I can't find sources explaining why you can't have an odd sided polygon like a triangle as quotient map for a surface.