Kawasaki's Theorem - Statement of The Theorem

Statement of The Theorem

Maekawa's theorem states that the number of mountain folds in a flat-folded vertex figure differs from the number of valley folds by exactly two folds. From this it follows that the total number of folds must be even. Therefore, suppose that a crease pattern consists of an even number 2n of creases radiating from a single vertex v, without specification of which creases should be mountain folds and which should be valley folds. In this crease pattern, let α₁, α₂, ⋯, α_2n be the consecutive angles between the creases around v, in clockwise order, starting at any one of the angles. Then Kawasaki's theorem is the statement that the crease pattern may be folded flat if and only if the alternating sum and difference of the angles adds to zero:

α₁ − α₂ + α₃ − ⋯ + α_{2n − 1} − α_2n = 0

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.