  We develop recursion equations to describe the three-dimensional shape of a
sheet upon which a series of concentric curved folds have been inscribed. In
the case of 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, we derive continuum equations, valid in the limit of
vanishing spacing between folds, to describe the smooth surface intersecting
all the mountain folds. We find that this surface has negative Gaussian
curvature with magnitude equal to the square of the fold's torsion. A series of
open folds with constant fold angle generate a helicoid.
