Mathematics of paper folding

In 1893, Indian civil servant T. Sundara Row published Geometric Exercises in Paper Folding which used paper folding to demonstrate proofs of geometrical constructions. This work was inspired by the use of origami in the kindergarten system. Row demonstrated an approximate trisection of angles and implied construction of a cube root was impossible.[4]

In 1922, Harry Houdini published "Houdini's Paper Magic," which described origami techniques that drew informally from mathematical approaches that were later formalized.[5]

The Beloch fold

In 1936 Margharita P. Beloch showed that use of the 'Beloch fold', later used in the sixth of the Huzita–Hatori axioms, allowed the general cubic equation to be solved using origami.[1]

In 1949, R C Yeates' book "Geometric Methods" described three allowed constructions corresponding to the first, second, and fifth of the Huzita–Hatori axioms.[6][7]

The Yoshizawa–Randlett system of instruction by diagram was introduced in 1961.[8]

Crease pattern for a Miura fold. The parallelograms of this example have 84° and 96° angles.

In 1980 was reported a construction which enabled an angle to be trisected. Trisections are impossible under Euclidean rules.[9]

Also in 1980, Kōryō Miura and Masamori Sakamaki demonstrated a novel map-folding technique whereby the folds are made in a prescribed parallelogram pattern, which allows the map to be expandable without any right-angle folds in the conventional manner. Their pattern allows the fold lines to be interdependent, and hence the map can be unpacked in one motion by pulling on its opposite ends, and likewise folded by pushing the two ends together. No unduly complicated series of movements are required, and folded Miura-ori can be packed into a very compact shape.[10] In 1985 Miura reported a method of packaging and deployment of large membranes in outer space,[11] and as early as 2012 this technique had been applied to solar panels on spacecraft.[12][13]

A diagram showing the first and last step of how origami can double the cube

In 1986, Messer reported a construction by which one could double the cube, which is impossible with Euclidean constructions.[14]

The first complete statement of the seven axioms of origami by French folder and mathematician Jacques Justin was written in 1986, but were overlooked until the first six were rediscovered by Humiaki Huzita in 1989.[15] The first International Meeting of Origami Science and Technology (now known as the International Conference on Origami in Science, Math, and Education) was held in 1989 in Ferrara, Italy. At this meeting, a construction was given by Scimemi for the regular heptagon.[16]

Around 1990, Robert J. Lang and others first attempted to write computer code that would solve origami problems.[17]

Mountain-valley counting

In 1996, Marshall Bern and Barry Hayes showed that the problem of assigning a crease pattern of mountain and valley folds in order to produce a flat origami structure starting from a flat sheet of paper is NP-complete.[18]

In 1999, a theorem due to Haga provided constructions used to divide the side of a square into rational fractions.[19][20]

In late 2001 and early 2002, Britney Gallivan proved the minimum length of paper necessary to fold it in half a certain number of times and folded a 4,000-foot-long (1,200 m) piece of toilet paper twelve times.[21][22]

In 2002, belcastro and Hull brought to the theoretical origami the language of affine transformations, with an extension from ${\displaystyle R}$² to ${\displaystyle R}$³ in only the case of single-vertex construction.[23]

In 2002, Alperin solved Alhazen's problem of spherical optics.[24] In the same paper, Alperin showed a construction for a regular heptagon.[24] In 2004, was proven algorithmically the fold pattern for a regular heptagon.[25] Bisections and trisections were used by Alperin in 2005 for the same construction.[26]

In 2003, Jeremy Gibbons, a researcher from the University of Oxford, described a style of functional programming in terms of origami. He coined this paradigm as "origami programming." He characterizes fold and unfolds as natural patterns of computation over recursive datatypes that can be framed in the context of origami.[27]

In 2005, principles and concepts from mathematical and computational origami were applied to solve Countdown, a game popularized in British television in which competitors used a list of source numbers to build an arithmetic expression as close to the target number as possible.[28]

In 2009, Alperin and Lang extended the theoretical origami to rational equations of arbitrary degree, with the concept of manifold creases.[29][30] This work was a formal extension of Lang's unpublished 2004 demonstration of angle quintisection.[30][31]

As a result of origami study through the application of geometric principles, methods such as Haga's theorem have allowed paperfolders to accurately fold the side of a square into thirds, fifths, sevenths, and ninths. Other theorems and methods have allowed paperfolders to get other shapes from a square, such as equilateral triangles, pentagons, hexagons, and special rectangles such as the golden rectangle and the silver rectangle. Methods for folding most regular polygons up to and including the regular 19-gon have been developed.[36] A regular n-gon can be constructed by paper folding if and only if n is a product of distinct Pierpont primes, powers of two, and powers of three.

Haga's theorems

BQ is always rational if AP is.

The side of a square can be divided at an arbitrary rational fraction in a variety of ways. Haga's theorems say that a particular set of constructions can be used for such divisions.[19][20] Surprisingly few folds are necessary to generate large odd fractions. For instance 1⁄5 can be generated with three folds; first halve a side, then use Haga's theorem twice to produce first 2⁄3 and then 1⁄5.

The accompanying diagram shows Haga's first theorem:

${\displaystyle BQ={\frac {2AP}{1+AP}}.}$

The function changing the length AP to QC is self inverse. Let x be AP then a number of other lengths are also rational functions of x. For example:

Haga's first theorem

AP	BQ	QC	AR	PQ
${\displaystyle x}$	${\displaystyle {\frac {2x}{1+x}}}$	${\displaystyle {\frac {1-x}{1+x}}}$	${\displaystyle {\frac {1-x^{2}}{2}}}$	${\displaystyle {\frac {1+x^{2}}{1+x}}}$
1⁄2	2⁄3	1⁄3	3⁄8	5⁄6
1⁄3	1⁄2	1⁄2	4⁄9	5⁄6
2⁄3	4⁄5	1⁄5	5⁄18	13⁄15
1⁄5	1⁄3	2⁄3	12⁄25	13⁄15

Doubling the cube

Doubling the cube: PB/PA = cube root of 2

The classical problem of doubling the cube can be solved using origami. This construction is due to Peter Messer:[38] A square of paper is first creased into three equal strips as shown in the diagram. Then the bottom edge is positioned so the corner point P is on the top edge and the crease mark on the edge meets the other crease mark Q. The length PB will then be the cube root of 2 times the length of AP.[14]

The edge with the crease mark is considered a marked straightedge, something which is not allowed in compass and straightedge constructions. Using a marked straightedge in this way is called a neusis construction in geometry.

Trisecting an angle

Trisecting the angle CAB

Angle trisection is another of the classical problems that cannot be solved using a compass and unmarked ruler but can be solved using origami.[39] This construction, which was reported in 1980, is due to Hisashi Abe.[38][9] The angle CAB is trisected by making folds PP' and QQ' parallel to the base with QQ' halfway in between. Then point P is folded over to lie on line AC and at the same time point A is made to lie on line QQ' at A'. The angle A'AB is one third of the original angle CAB. This is because PAQ, A'AQ and A'AR are three congruent triangles. Aligning the two points on the two lines is another neusis construction as in the solution to doubling the cube.[40][9]

The problem of rigid origami, treating the folds as hinges joining two flat, rigid surfaces, such as sheet metal, has great practical importance. For example, the Miura map fold is a rigid fold that has been used to deploy large solar panel arrays for space satellites.

The napkin folding problem is the problem of whether a square or rectangle of paper can be folded so the perimeter of the flat figure is greater than that of the original square.

The placement of a point on a curved fold in the pattern may require the solution of elliptic integrals. Curved origami allows the paper to form developable surfaces that are not flat.[41] Wet-folding origami is a technique evolved by Yoshizawa that allows curved folds to create an even greater range of shapes of higher order complexity.

The maximum number of times an incompressible material can be folded has been derived. With each fold a certain amount of paper is lost to potential folding. The loss function for folding paper in half in a single direction was given to be ${\displaystyle L={\tfrac {\pi t}{6}}(2^{n}+4)(2^{n}-1)}$, where L is the minimum length of the paper (or other material), t is the material's thickness, and n is the number of folds possible.[42] The distances L and t must be expressed in the same units, such as inches. This result was derived by Britney Gallivan, a high schooler from California, in December 2001. In January 2002, she folded a 4,000-foot-long (1,200 m) piece of toilet paper twelve times in the same direction, debunking a long-standing myth that paper cannot be folded in half more than eight times.[21][22]

The fold-and-cut problem asks what shapes can be obtained by folding a piece of paper flat, and making a single straight complete cut. The solution, known as the fold-and-cut theorem, states that any shape with straight sides can be obtained.

A practical problem is how to fold a map so that it may be manipulated with minimal effort or movements. The Miura fold is a solution to the problem, and several others have been proposed.[43]

Computational origami is a branch of computer science that is concerned with studying algorithms for solving paper-folding problems. In the early 1990s, origamists participated in a series of origami contests called the Bug Wars in which artists attempted to out-compete their peers by adding complexity to their origami bugs. Most competitors in the contest belonged to the Origami Detectives, a group of acclaimed Japanese artists.[44] Robert Lang, a research-scientist from Stanford University and the California Institute of Technology, also participated in the contest. The contest helped initialize a collective interest in developing universal models and tools to aid in origami design and foldability.[44]

Software & tools

Animation of folds to make a Samurai helmet, also called a kabuto. (On a laptop computer, Julia and GLMakie generated[49] the 66 second .mp4 video in 10 seconds.)

There are several software design tools that are used for origami design. Users specify the desired shape or functionality and the software tool constructs the fold pattern and/or 2D or 3D model of the result. Researchers at the Massachusetts Institute of Technology, Georgia Tech, University of California Irvine, University of Tsukuba, and University of Tokyo have developed and posted publicly available tools in computational origami. TreeMaker, ReferenceFinder, OrigamiDraw, and Origamizer are among the tools that have been used in origami design.[50]

There are other software solutions associated with building computational origami models using non-paper materials such as Cadnano in DNA origami.[51]

Wikimedia Commons has media related to Origami mathematics.

• Flexagon
• Lill's method
• Napkin folding problem
• Map folding
• Regular paperfolding sequence (for example, the dragon curve)

• Demaine, Erik D., "Folding and Unfolding", PhD thesis, Department of Computer Science, University of Waterloo, 2001.
• Friedman, Michael (2018). A History of Folding in Mathematics: Mathematizing the Margins. Science Networks. Historical Studies. Vol. 59. Birkhäuser. doi:10.1007/978-3-319-72487-4. ISBN 978-3-319-72486-7.
• Geretschlager, Robert (1995). "Euclidean Constructions and the Geometry of Origami". Mathematics Magazine. 68 (5): 357–371. doi:10.2307/2690924. JSTOR 2690924.
• Haga, Kazuo (2008). Fonacier, Josefina C; Isoda, Masami (eds.). Origamics: Mathematical Explorations Through Paper Folding. University of Tsukuba, Japan: World Scientific Publishing. ISBN 978-981-283-490-4.
• Lang, Robert J. (2003). Origami Design Secrets: Mathematical Methods for an Ancient Art. A K Peters. ISBN 978-1-56881-194-9.
• Dureisseix, David, "Folding optimal polygons from squares", Mathematics Magazine 79(4): 272–280, 2006. doi:10.2307/27642951
• Dureisseix, David, "An Overview of Mechanisms and Patterns with Origami", International Journal of Space Structures 27(1): 1–14, 2012. doi:10.1260/0266-3511.27.1.1

Wikimedia Commons has media related to Origami mathematics.

• Dr. Tom Hull. "Origami Mathematics Page".
• Paper Folding Geometry at cut-the-knot
• Dividing a Segment into Equal Parts by Paper Folding at cut-the-knot
• Britney Gallivan has solved the Paper Folding Problem
• Overview of Origami Axioms
• Introduction to Statistics with Origami by Mario Cigada