We develop equations that describe the three-dimensional shape of a sheet upon which a series of concentric, curved folds have been inscribed. When there is no stretching outside the fold, the three-dimensional shape of a single fold prescribes the shape of the entire origami structure. To better explore these structures and their mechanical properties, we develop a description of the smooth surface intersecting all the mountain folds. We find that this surface is always saddle-shaped. A series of open, circular folds (as seen on the right figure) with constant fold angle generate a classical minimal surface known as the helicoid.