Maekawa's theorem is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

in the

mathematics of paper folding The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper f ...

named after

Jun Maekawa is a Japanese software engineer, mathematician, and origami artist. He is known for popularizing the method of utilizing crease patterns in designing origami models, with his 1985 publication ''Viva Origami'', as well as other paperfolding-related ...

. It relates to flat-foldable

origami ) is the Japanese paper art, art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of pape ...

crease pattern A crease pattern is an origami diagram that consists of all or most of the creases in the final model, rendered into one image. This is useful for diagramming complex and super-complex models, where the model is often not simple enough to diagram e ...

s and states that at every

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

, the numbers of valley and mountain folds always differ by two in either direction. The same result was also discovered by Jacques Justin and, even earlier, by S. Murata.

Parity and coloring

One consequence of Maekawa's theorem is that the total number of folds at each vertex must be an

even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...

. This implies (via a form of planar graph duality between

Eulerian graph In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends ...

s and

bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are ...

s) that, for any flat-foldable crease pattern, it is always possible to

color Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associ ...

the regions between the creases with two colors, such that each crease separates regions of differing colors.. See in particular Theorem 3.1 and Corollary 3.2. The same result can also be seen by considering which side of the sheet of paper is uppermost in each region of the folded shape.

Related results

Maekawa's theorem does not completely characterize the flat-foldable vertices, because it only takes into account the numbers of folds of each type, and not their angles. Kawasaki's theorem gives a complementary condition on the angles between the folds at a vertex (regardless of which folds are mountain folds and which are valley folds) that is also necessary for a vertex to be flat-foldable.

References

External links

* {{Mathematics of paper folding Paper folding Recreational mathematics