Mathematics of Origami: Flat-Folding

                 Mathematics of Origami

Think about it, mathematics combined with the art of origami. Well, it's a real thing! The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper's flat-foldability( whether the paper model can be flattened without damaging it), and the use of paper folding to solve cubic equations. Well, in this last post of the blog on origami, let's dive into the mathematics of flat- folding.

-----> FLAT-FOLDING

The construction of origami models is sometimes displayed as crease patterns. The major question about such creases is whether a given crease pattern can be folded to a flat model, and if so, how to fold them; this is known as an NP-Complete Problem. There are 3 mathematical rules for producing flat-foldable crease patterns:

1. MAEKAWA'S THEOREM:  At any vertex, the number of valley folds and mountain folds always differ by 2.  Every vertex has an even number of creases,  and therefore also the region between the creases can be colored with 2 colors.

 In this one-vertex crease pattern, the number of mountain folds( the 5 folds with colored side out) differs by 2 from the number of valley folds(the 3 folds with white side out).

Paper exhibits zero Gaussian curvature at all points on its surface, and only folds naturally along lines of zero curvature. Curved surfaces that can't be flattened can be produced using a non-folded crease in the paper, as it is easily done with wet paper or your fingernail.

2. KAWASAKI'S THEOREM: Also known as the Kawasaki-Justin theorem, it states that the crease pattern is flatly foldable only if adding and subtracting the angles of consecutive folds around the vertex, alternatingly, gives an alternating sum of 0. Crease patterns with more than 1 vertex do not obey such a criterion and are NP-Hard to fold. This theorem is named after one of its discoverers, Toshikazu Kawasaki, who is popular for his origami roses and more amazing work. Jacques Justin and Koji Husimi also contributed.

In this example, the alternating sum of angles( clockwise from the bottom) is 90°- 45° + 22.5° - 22.5° + 45° - 90° + 22.5° - 22.5°= 0° . Since it sums up to 0, the crease pattern is flatly foldable.