Maekawa's theorem

The set of Maekawa is a mathematical theorem about origami.

Statement

The statement of the theorem by Maekawa refers to folded flat characters whose folds converge in the unfolded state in a center. You share all the wrinkles of such flat-folded figure in mountain folds (hereinafter referred to as, of English "mountain " = mountain) and valley folds (hereinafter referred to, of English " valley" = valley ) a. This classification can be done in two ways: either one considers the wrinkles with outward facing angles as mountain folds and wrinkles with inward facing angles as valley folds, or vice versa.

Looking at the folds with outward facing angles as mountain folds, so says the set of Maekawa:

In the other case, he says:

Conclusion

In both cases,

Thus, divisible by two, and also by the addition of, the total number of folds. So that the total number of folds is straight.

Idea of ​​proof

First one chooses the mountain folds as the folds expand to the outside, ie they have an internal angle of (since the figure is folded flat). Is the total number of folds of the figure. Summing up by the Centre in which to meet all the wrinkles, remote side as a polygon with corners so each corner corresponds to a fold, then: The sum of the interior angles of this polygon is the same. But if one has the mountain folds chosen as the outward-pointing angle so the valley folds are inwardly facing angle, that have an internal angle of (since the figure is folded flat). Thus, the interior angle sum is also equal. Since the interior angle sum can not take two different values, that is true. It follows by equivalence transformations.

If mountain folds and valley folds is defined vice versa, exchanging the roles and it is.