Kawasaki's theorem or Kawasaki–Justin theorem is a theorem in the mathematics of paper folding that describes the crease patterns with a single vertex that may be folded to form a flat figure. It states that the pattern is flat-foldable if and only if alternatingly adding and subtracting the angles of consecutive folds around the vertex gives an alternating sum of zero. Crease patterns with more than one vertex do not obey such a simple criterion, and are NP-hard to fold. The theorem is named after one of its discoverers, Toshikazu Kawasaki. However, several others also contributed to its discovery, and it is sometimes called the Kawasaki–Justin theorem or Husimi's theorem after other contributors, Jacques Justin and Kôdi Husimi. Statement A one-vertex crease pattern consists of a set of rays or creases drawn on a flat sheet of paper, all emanating from the same point interior to the sheet. (This point is called the vertex of the pattern.) Each crease must be folded, but the pattern does not specify whether the folds should be mountain folds or valley folds. The goal is to determine whether it is possible to fold the paper so that every crease is folded, no folds occur elsewhere, and the whole folded sheet of paper lies flat. To fold flat, the number of creases must be even. This follows, for instance, from Maekawa's theorem, which states that the number of mountain folds at a flat-folded vertex differs from the number of valley folds by exactly two folds. Therefore, suppose that a crease pattern consists of an even number of creases, and let be the consecutive angles between the creases around the vertex, in clockwise order, starting at any one of the angles. Then Kawasaki's theorem states that the crease pattern may be folded flat if and only if the alternating sum and difference of the angles adds to zero: An equivalent way of stating the same condition is that, if the angles are partitioned into two alternating subsets, then the sum of the angles in either of the two subsets is exactly 180 degrees. However, this equivalent form applies only to a crease pattern on a flat piece of paper, whereas the alternating sum form of the condition remains valid for crease patterns on conical sheets of paper with nonzero defect at the vertex. Local and global flat-foldability Kawasaki's theorem, applied to each of the vertices of