Mathematics Of Paper Folding

Pure origami

Flat folding

Two-colorability.

Mountain-valley counting.

Angles around a vertex.

The construction of origami models is sometimes shown as crease patterns. The major questions about such crease patterns are whether a crease pattern can be folded to a flat model, and whether deciding how to fold them is an NP problem. Related problems when the creases are orthogonal are called map folding problems. There are four mathematical rules for producing flat-foldable origami crease patterns:

crease patterns are two colorable

at any vertex the number of valley and mountain folds always differ by two in either direction

Kawasaki’s theorem: at any vertex, the sum of all the odd angles adds up to 180 degrees, as do the even.

a sheet can never penetrate a fold.

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

Assigning a crease pattern mountain and valley folds in order to produce a flat model has been proven by Marshall Bern and Barry Hayes to be NP complete. Further references and technical results are discussed in Part II of Geometric Folding Algorithms.

Axioms

Some classical construction problems of geometry namely trisecting an arbitrary angle, or doubling the cube are proven to be unsolvable using compass and straightedge, but can be solved using only a few paper folds. Paper fold strips can be constructed to solve equations up to degree 4. (The Huzitaatori axioms are one important contribution to this field of study.) Complete methods for solving all equations up to degree 4 by applying such methods are discussed in detail in Geometric Origami.

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.

Constructions

Haga’s theorems

BQ is always a 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.. Surprisingly few folds are necessary to generate large odd fractions. For instance 15 can be generated with three folds; first halve a side, then use Haga’s theorem twice to produce first 23 and then 15.

The accompanying diagram shows Haga’s first theorem:

Interestingly 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

x

12

23

13

38

56

13

12

12

49

56

23

45

15

518

1315

15

13

23

1225

1315

Doubling the cube

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

The classical problem of doubling the cube can be solved by first creasing a square of paper 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.

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.

Related problems

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.

Curved origami also poses a (very different) set of mathematical challenges. Curved origami allows the paper to form developable surfaces that are not flat.

Wet-folding origami allows an even greater range of shapes.

The loss function for folding paper in half in a single direction was given to be , 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. This function was given by Britney Gallivan in 2001 (then only a high school student) who managed to fold a sheet of paper in half 12 times, contrary to the popular belief that paper of any size could be folded at most eight times.

See also

Napkin folding problem

Notes

^ Thomas C. Hull (2002). “The Combinatorics of Flat Folds: a Survey”. The Proceedings of the Third International Meeting of Origami Science, Mathematics, and Education. AK Peters. ISBN 9781568811819. http://kahuna.merrimack.edu/~thull/papers/flatsurvey.pdf. 

^ TED.com

^ The Complexity of Flat Origami

^ Demaine, Erik; O’Rourke, Joseph (July 2007), Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, ISBN 978-0-521-85757-4, http://www.gfalop.org 

^ Origami Geometric Constructions

^ a b Geretschlger, Robert (2008). Geometric Origami. UK: Arbelos. ISBN 978-0-9555477-1-3. http://www.arbelos.co.uk/GeometricOrigami.html. 

^ Koshiro. “How to Divide the Side of Square Paper”. Japan Origami Academic Society. http://origami.gr.jp/Archives/People/CAGE_/divide/02-e.html. 

^ Peter Messer (1986). “Problem 1054”. Crux Mathematicorum 12 (10): pp. 284285.. 

^ Siggraph: “Curved Origami”

^ Weisstein, Eric W., “Folding” from MathWorld.

Further reading

Robert J. Lang (2003). Origami Design Secrets: Mathematical Methods for an Ancient Art. A K Peters. ISBN 1568811942. 

Kazuo Haga (2008). Josefina C Fonacier; Masami Isoda. eds. Origamics: Mathematical Explorations through Paper Folding. World Scientific. ISBN 9789812834898. 

Haga, Kazuo (2008), Fonacier, Josefina C; Isoda, Masami, eds., Origamics: Mathematical Explorations Through Paper Folding, University of Tsukuba, Japan: World Scientific Publishing, ISBN 978-9812834904 

External links

Origami portal

Dr. Tom Hull, Origami Mathematics Page, http://mars.wnec.edu/~th297133/origamimath.html 

Origami & Math by Eric M. Andersen

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

Folding Paper – Great Moments in Science – ABC

Origami & geometry in English and in French

Origami & geometry in English and in Hebrew

Origami & geometry in Spanish

Categories: Recreational mathematics | Paper folding | Origami