Consider the map $\varphi : \mathbb R^2 \to \mathbb R^2$ defined by $(x,y) \mapsto (x,y^2)$.
Apparently this map is called a fold as the $(x,y)$-plane is folded over and creased along the axis $y=0$.
But I really don't see how this is the case: $y^2$ does not "fold" or "crease" anything it just bends the plane very very slightly. No?
Please could someone explain to me how this map folds the plane? I obviously misunderstand it completely.