I did a bit of this at uni.. non-Euclidean geometries are based on non-flat surfaces Riemann and Lobachevsky both came up with kind of convex/concave non-euclidean geometries if you imagine a triangle on a curved surface, the angles are exaggerated and can add up to more or less than 180 Euclid's parallel postulate was one of the axioms that he used to prove his geometry assuming his axioms to be true, his proof was correct but if you do not assume it, the non-Euclidean geometries become consistent theories. people have tried and failed to prove the parallel postulate is true i think the whole curved surface thing ties in with Einstein.. :zzz: