an arbitrary crease pattern, determines whether the crease pattern is locally flat-foldable, meaning that the part of the crease pattern near the vertex can be flat-folded. However, there exist crease patterns that are locally flat-foldable but that have no global flat folding that works for the whole crease pattern at once. conjectured that global flat-foldability could be tested by checking Kawasaki's theorem at each vertex of a crease pattern, and then also testing bipartiteness of an undirected graph associated with the crease pattern. However, this conjecture was disproven by , who showed that Hull's conditions are not sufficient. More strongly, Bern and Hayes showed that the problem of testing global flat-foldability is NP-complete. Proof To show that Kawasaki's condition necessarily holds for any flat-folded figure, it suffices to observe that, at each fold, the orientation of the paper is reversed. Thus, if the first crease in the flat-folded figure is placed in the plane parallel to the -axis, the next crease must be rotated from it by an angle of , the crease after that by an angle of (because the second angle has the reverse orientation from the first), etc. In order for the paper to meet back up with itself at the final angle, Kawasaki's condition must be met. Showing that the condition is also a sufficient condition is a matter of describing how to fold a given crease pattern so that it folds flat. That is, one must show how to choose whether to make mountain or valley folds, and in what order the flaps of paper should be arranged on top of each other. One way to do this is to choose a number such that the partial alternating sum is as small as possible. Either and the partial sum is an empty sum that is also zero, or for some nonzero choice of the partial sum is negative. Then, accordion fold the pattern, starting with angle and alternating between mountain and valley folds, placing each angular wedge of the paper below the previous folds. At each step until the final fold, an accordion fold of this type will never self-intersect. The choice of ensures that the first wedge sticks out to the left of all the other folded pieces of paper, allowing the final wedge to connect back up to it. An alternative proof of sufficiency can be used to show that there are many