pattern meeting the conditions of the theorem to independent work in the early 1990s by Azuma, Justin, and Ewins and Hull. Although Kawasaki's theorem completely describes the folding patterns that have flat-folded states, it does not describe the folding process needed to reach that state. For some (multi-vertex) folding patterns, it is necessary to curve or bend the paper while transforming it from a flat sheet to its flat-folded state, rather than keeping the rest of the paper flat and only changing the dihedral angles at each fold. For rigid origami (a type of folding that keeps the surface flat except at its folds, suitable for hinged panels of rigid material rather than flexible paper), the condition of Kawasaki's theorem turns out to be sufficient for a single-vertex crease pattern to move from an unfolded state to a flat-folded state. References External links Mathematical theorems Paper folding