Kawasaki's theorem

The set of Kawasaki is a formulated by Toshikazu Kawasaki mathematical theorem about origami.

Statement

A folding pattern with a center in which meet all the wrinkles, you can flat fold (ie, flatten ) if and only if the angle between two successive folding of the folding pattern ( in the unfolded state ) result in the alternating sum zero, ie:

The angle between two successive folds of the fold pattern are (the number of wrinkles is by the set of Maekawa even).

Alternative formulation

Since for each crease pattern on a flat paper with a center in which meet all the wrinkles, then: is obtained (if one measures the angle in radians ) by the addition method:

Since the equivalence is transitive, we can rephrase the sentence as follows in the case of a flat paper:

A folding pattern with a center in which meet all the wrinkles, you can fold flat (ie, flatten ) if and only if:

Wherein the angle between two successive folds of the fold pattern.

For a non - flat output paper this equivalence does not hold, that is, the alternative formulation is not applicable. Nevertheless, the first formulation in non- flat output papers, such as cones, but applicable.

Idea of ​​proof

The proof is divided into two parts, namely, return the direction of the equivalence:

• A flat-folded figure Imagine a center in which meet all the wrinkles, before. Be this figure in flat folded state. If you walk along the edge exactly once, so the corresponding position on the edge with an angle at the center. It now records all angular differences between two folds and numbers them chronologically by date of Abschreitens. If you have completed the entire distance, one obtains the angle. As one with a folding changes direction, and at the end is again at exactly the same angle, where it was started walking (with respect to the center, in the folded state ), applies.

• Be for a figure. Now choose a, so is minimal ( this expression must then be less than or equal to zero, because otherwise a small expression of this kind ). Then you fold from the angle of the figure to the accordion - type, ie alternately corresponds first to the left and then to the right, so right negative angles. Considering this, the position of the current fold, so this never shifts to the right, because at first angles, the sum of the angle increases (or remains the same ) to catch up to zero (or retain the value ). In the following, the current angles fold then never wander further to the right than the first fold at the angle at which you started, otherwise a smaller sum would exist. As the number of wrinkles is even, and is due to the commutative law of addition: