Maekawa's theorem

The set of Maekawa is a mathematical theorem about Origami .

statement

The statement in Maekawa's theorem relates to figures folded flat, the folds of which converge in a center when unfolded. All folds of such a flat folded figure are divided into mountain folds (hereinafter referred to as, from English "mountain" = mountain) and valley folds (hereinafter referred to as, from English "valley" = valley). This classification can be done in two ways: Either one considers the folds with angles pointing outwards as mountain folds and the folds with angles pointing inwards as valley folds, or vice versa. ${\displaystyle M}$ ${\displaystyle V}$

If one considers the folds with outward-pointing angles as mountain folds, then Maekawa's theorem says:

${\displaystyle M-V=2}$

In the other case it says:

${\displaystyle M-V=-2}$

Inference

The following applies in both cases:

${\displaystyle |M-V|=2}$

So is divisible by two, and by adding also , where is the total number of folds. So the total number of folds is even. ${\displaystyle M-V}$ ${\displaystyle 2\cdot V}$ ${\displaystyle n=M+V}$ ${\displaystyle n}$

Proof idea

First, the mountain folds are chosen as the folds that point outwards, i.e. i.e., they have an interior angle of (since the figure is folded flat). Let be the total number of folds on the figure. If the side remote from the center, where all the folds meet, is understood as a polygon with corners, so that each corner corresponds to a fold, then the following applies: The sum of the interior angles of this polygon is the same . But if you have chosen the mountain folds as angles pointing outwards, then the valley folds are angles pointing inwards, i.e. have an interior angle of (since the figure is folded flat). This means that the sum of the interior angles is also the same . Since the sum of the interior angles cannot have two different values, the following applies . It follows by equivalence transformations : . ${\displaystyle 0^{\circ }}$ ${\displaystyle n=M+V}$ ${\displaystyle n}$ ${\displaystyle (n-2)\cdot 180^{\circ }}$ ${\displaystyle 360^{\circ }}$ ${\displaystyle 0^{\circ }\cdot M+360^{\circ }\cdot V}$ ${\displaystyle 0^{\circ }\cdot M+360^{\circ }\cdot V=(n-2)\cdot 180^{\circ }=(M+V-2)\cdot 180^{\circ }}$ ${\displaystyle M-V=2}$

If one mountain folds and valley folds defined vice versa, exchange and the roles and it applies . ${\displaystyle M}$ ${\displaystyle V}$ ${\displaystyle V-M=2\Leftrightarrow M-V=-2}$

Individual evidence

1. ^ ^{A b} Thomas C. Hull: The Combinatorics of Flat Folds: a Survey. 2002, accessed May 22, 2013 . 
2. ↑ Joseph M. Kudrle: Origami & Mathematics. ( MS PowerPoint ; 3.7 MB) Retrieved May 22, 2013 .