View source for Mathematics of paper folding

{{short description|Overview of the mathematics of paper folding}}
[[File:MapFoldings-2x2.png|thumb|upright=1.2|[[Map folding]] for a 2&times;2 grid of squares: there are eight different ways to fold such a map along its creases]]
The discipline of [[origami]] or paper folding has received a considerable amount of [[mathematics|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 folds to solve [[equation|mathematical equations]] up to the third order.<ref name=hull11/>

Computational origami is a recent branch of [[computer science]] that is concerned with studying algorithms that solve paper-folding problems. The field of computational origami has also grown significantly since its inception in the 1990s with Robert Lang's TreeMaker algorithm to assist in the precise folding of bases.<ref>{{Cite news |title=origami - History of origami {{!}} Britannica |url=https://www.britannica.com/art/origami/History-of-origami |access-date=2022-05-08 |newspaper=Encyclopedia Britannica |language=en}}</ref> Computational origami results either address origami design or origami foldability.<ref name=":0">"Lecture: Recent Results in Computational Origami". ''Origami USA: We are the American national society devoted to origami, the art of paperfolding''. Retrieved 2022-05-08.</ref> In origami design problems, the goal is to design an object that can be folded out of paper given a specific target configuration. In origami foldability problems, the goal is to fold something using the creases of an initial configuration. Results in origami design problems have been more accessible than in origami foldability problems.<ref name=":0" />

==History==
{{see also|History of origami}}
In 1893, Indian [[civil service|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.<ref name=rao17>{{cite book |title=Geometric Exercises in Paper Folding|title-link= Geometric Exercises in Paper Folding |author= T. Sundara Row   |publisher=[[The Open Court Publishing Company]] |year=1917 |editor1-last=Beman |editor1-first=Wooster |editor2-last=Smith |editor2-first=David}}</ref>

In 1922, [[Harry Houdini]] published "Houdini's Paper Magic," which described origami techniques that drew informally from mathematical approaches that were later formalized.<ref>{{Cite book |last=Houdini |first=Harry|author-link=Harry Houdini |url=https://en.wikisource.org/wiki/Houdini's_Paper_Magic |title=Houdini's Paper Magic}}</ref>

[[File:Group1-Origami The Beloch Fold.png|thumb|right|250px|The Beloch fold]]
In 1936 [[Margherita Piazzola Beloch|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.<ref name=hull11>{{cite journal
 | last = Hull | first = Thomas C. | author-link = Tom Hull (mathematician)
 | doi = 10.4169/amer.math.monthly.118.04.307
 | issue = 4
 | journal = American Mathematical Monthly
 | mr = 2800341
 | pages = 307–315
 | title = Solving cubics with creases: the work of Beloch and Lill
 | url = http://mars.wne.edu/~thull/papers/amer.math.monthly.118.04.307-hull.pdf
 | volume = 118
 | year = 2011| s2cid = 2540978
 }}</ref>

In 1949, R C Yeates' book "Geometric Methods" described three allowed constructions corresponding to the first, second, and fifth of the Huzita–Hatori axioms.<ref>{{cite book |title=Geometric constructions |title-link=Geometric Constructions|author=George Edward Martin |publisher=Springer |year=1997 |isbn= 978-0-387-98276-2 |page=145}}</ref><ref>{{cite book|title=Geometric Tools |author=Robert Carl Yeates |publisher= Louisiana State University  |year=1949}}</ref>

The [[Yoshizawa–Randlett system]] of instruction by diagram was introduced in 1961.<ref name=obible>{{cite book |title=The Origami Bible |author=Nick Robinson |page=18 |year=2004 |publisher=Chrysalis Books |isbn=978-1-84340-105-6}}</ref>

[[File:Miura-Ori CP.svg|thumb|250px|[[Crease pattern]] for a Miura fold. The parallelograms of this example have 84° and 96° angles.]]
In 1980 a construction was reported which enabled an angle to be trisected. Trisections are impossible under Euclidean rules.<ref name=hullcomp/>

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.<ref name=bain80>{{citation|url=https://www.britishorigami.info/academic/mathematics/the-miura-ori-map/|title=The Miura-Ori map|first=Ian|last=Bain|journal=[[New Scientist]]|year=1980}}. Reproduced in ''British Origami'', 1981, and online at the British Origami Society web site.</ref> In 1985 Miura reported a method of packaging and deployment of large membranes in outer space,<ref name=miura85>{{citation
 | last = Miura | first = K.
 | publisher = The Institute of Space and Astronautical Science
 | series = Tech. Report 618
 | title = Method of packaging and deployment of large membranes in space
 | year = 1985}}</ref> and as early as 2012 this technique had been applied to [[solar panels on spacecraft]].<ref name=jaxa>{{cite news |publisher=Japan Aerospace Exploration Agency|archive-url=https://web.archive.org/web/20051125174630/http://www.isas.jaxa.jp/e/enterp/missions/complate/sfu/2dsa.shtml |url=http://www.isas.jaxa.jp/e/enterp/missions/complate/sfu/2dsa.shtml |archive-date=25 November 2005|title=2D Array}}</ref><ref name=yutaka12>{{citation|first=Yutaka|last=Nishiyama|authorlink=Yutaka Nishiyama|url=http://www.ijpam.eu/contents/2012-79-2/8/8.pdf|title=Miura folding: Applying origami to space exploration|journal=International Journal of Pure and Applied Mathematics|volume=79|issue=2|pages=269–279|year=2012}}</ref>

[[File:Group1-Origami Doubling the Cube .png|thumb|right|250px|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.<ref name=messer86/>

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.<ref name=jjhh89>Justin, Jacques, "Resolution par le pliage de l'equation du troisieme degre et applications geometriques", reprinted in ''Proceedings of the First International Meeting of Origami Science and Technology'', H. Huzita ed. (1989), pp. 251–261.</ref> 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]].<ref name=scimemi90>Benedetto Scimemi, Regular Heptagon by Folding, Proceedings of Origami, Science and Technology, ed. H. Huzita., Ferrara, Italy, 1990</ref>

Around 1990, [[Robert J. Lang]] and others first attempted to write computer code that would solve origami problems.<ref name="lnpow">{{cite news |last1=Newton |first1=Liz |title=The power of origami |url=https://plus.maths.org/content/power-origami |agency=+ plus magazine |publisher=University of Cambridge |date=1 December 2009}}</ref>

[[File:Maekawas Theorem.svg|thumb|100px|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]].<ref name=bern96/>

In 1999, a theorem due to Haga provided constructions used to divide the side of a square into rational fractions.<ref name=howto/><ref name=haga99/>

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 {{convert|4000|ft|m|adj=mid|-long}} piece of toilet paper twelve times.<ref name="Mathworld"/><ref name="d'Agostino"/>

In 2002, [[Sarah-Marie Belcastro]] and [[Tom Hull (mathematician)|Tom Hull]] brought to the theoretical origami the language of [[affine transformations]], with an extension from <math>R</math><sup>2</sup> to <math>R</math><sup>3</sup> in only the case of single-vertex construction.<ref name="belcastro02">{{cite journal |doi=10.1016/S0024-3795(01)00608-5|title=Modelling the folding of paper into three dimensions using affine transformations|year=2002|last1=Belcastro|first1=Sarah-Marie|author1-link=Sarah-Marie Belcastro|last2=Hull|first2=Thomas C.|author2-link=Tom Hull (mathematician)|journal=Linear Algebra and Its Applications|volume=348|issue=1–3|pages=273–282|doi-access=free}}</ref>

<!--An origamic view of Alhazen’s optical problem, in Proceedings of O3-Science, Education and Mathematics, T. Hull, ed., A. K. Peters, Natick, MA, 2002, pp. 83–93.-->
In 2002, Alperin solved [[Alhazen's problem]] of spherical optics.<ref name="alperin02">{{cite book |doi=10.1201/b15735 |last1=Alperin|first1=Roger C.|title=Mathematical Origami: Another View of Alhazen's Optical Problem |chapter=Ch.12 |pages=83–93 |date=2002 |isbn=9780429064906 |editor1-last=Hull |editor1-first=Thomas }}</ref> In the same paper, Alperin showed a construction for a regular heptagon.<ref name=alperin02/> In 2004, was proven algorithmically the fold pattern for a regular heptagon.<ref name="robu04">{{cite book |doi=10.1007/11615798_2|chapter=Computational Origami Construction of a Regular Heptagon with Automated Proof of Its Correctness|title=Automated Deduction in Geometry|series=Lecture Notes in Computer Science|year=2006|last1=Robu|first1=Judit|last2=Ida|first2=Tetsuo|last3=Ţepeneu|first3=Dorin|last4=Takahashi|first4=Hidekazu|last5=Buchberger|first5=Bruno|volume=3763|pages=19–33|isbn=978-3-540-31332-8}}</ref> Bisections and trisections were used by Alperin in 2005 for the same construction.<ref name="alperin05">{{cite journal |doi=10.2307/30037438|jstor=30037438|last1=Alperin|first1=Roger C.|title=Trisections and Totally Real Origami|journal=The American Mathematical Monthly|year=2005|volume=112|issue=3|pages=200–211|arxiv=math/0408159}}</ref>

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.<ref>{{Cite web |last=Gibbons |first=Jeremy |date=2003 |title=Origami Programming |url=https://www.cs.ox.ac.uk/jeremy.gibbons/publications/origami.pdf}}</ref>

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.<ref>{{Cite journal |last1=Bird |first1=Richard |last2=Mu |first2=Shin-Cheng |date=September 2005 |title=Countdown: A case study in origami programming |journal=Journal of Functional Programming |language=en |volume=15 |issue=5 |pages=679–702 |doi=10.1017/S0956796805005642 |doi-broken-date=1 November 2024 |s2cid=46359986 |issn=1469-7653|doi-access=free }}</ref>

In 2009, Alperin and Lang extended the theoretical origami to rational equations of arbitrary degree, with the concept of manifold creases.<ref name="alperin09">{{cite conference |last1=Lang |first1=Robert J. |last2=Alperin |first2=Roger C. |title=Origami 4 |chapter=One-, Two-, and Multi-Fold Origami Axioms |journal=Origami<sup>4</sup>: Fourth International Meeting of Origami Science, Mathematics, and Education |date=2009 |pages=383–406 |doi=10.1201/b10653-38 |isbn=9780429106613 |chapter-url=http://pub.math.leidenuniv.nl/~edixhovensj/talks/2017/2017_03_27origami_LiOdag/o4_multifold_axioms.pdf}}</ref><ref name="ugmo">{{cite book |last1=Bertschinger |first1=Thomas H. |last2=Slote |first2=Joseph |last3=Spencer |first3=Olivia Claire |last4=Vinitsky |first4=Samuel |title=The Mathematics of Origami |publisher=Carleton College |url=https://joeslote.com/documents/origami_undergrad_thesis.pdf}}</ref> This work was a formal extension of Lang's unpublished 2004 demonstration of angle quintisection.<ref name=ugmo/><ref name="quinti">{{cite web |last1=Lang |first1=Robert J. |title=Angle Quintisection |url=https://langorigami.com/wp-content/uploads/2015/09/quintisection.pdf |website=langorigami.com |access-date=16 January 2021 |date=2004}}</ref>

==Pure origami==

===Flat folding===
[[File:Lang rule one.png|thumb|100px|Two-colorability]]
[[File:Lang rule three.png|thumb|100px|Angles around a vertex]]

The construction of origami models is sometimes shown as crease patterns. The major question about such crease patterns is whether a given crease pattern can be folded to a flat model, and if so, how to fold them; this is an [[NP complete|NP-complete problem]].<ref>{{cite book |year=2002 |publisher=AK Peters |isbn=978-1-56881-181-9
|title=The Proceedings of the Third International Meeting of Origami Science, Mathematics, and Education
|author=Thomas C. Hull |chapter=The Combinatorics of Flat Folds: a Survey
|arxiv=1307.1065 }}</ref> Related problems when the creases are orthogonal are called [[map folding]] problems. There are three mathematical rules for producing flat-foldable origami [[crease pattern]]s:<ref>{{cite web | url=http://www.ted.com/index.php/talks/robert_lang_folds_way_new_origami.html | title=Robert Lang folds way-new origami}}</ref>

# [[Maekawa's theorem]]: at any vertex the number of valley and mountain folds always differ by two.
#: It follows from this that every vertex has an even number of creases, and therefore also the regions between the creases can be colored with two colors.
# [[Kawasaki's theorem]] or Kawasaki-Justin theorem: at any vertex, the sum of all the odd angles (see image) 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 that 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]].<ref name=bern96>{{cite conference
 | last1 = Bern | first1 = Marshall
 | last2 = Hayes | first2 = Barry
 | contribution = The complexity of flat origami
 | contribution-url = http://dl.acm.org/citation.cfm?id=313918
 | mr = 1381938
 | pages = 175–183
 | publisher = ACM, New York
 | title = Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (Atlanta, GA, 1996)
 | year = 1996}}</ref> Further references and technical results are discussed in Part II of ''[[Geometric Folding Algorithms]]''.<ref name="GFALOP">{{cite book
 | last1 = Demaine | first1 = Erik D. | author1-link = Erik Demaine
 | last2 = O'Rourke | first2 = Joseph | author2-link = Joseph O'Rourke (professor)
 | doi = 10.1017/CBO9780511735172
 | isbn = 978-0-521-85757-4
 | mr = 2354878
 | publisher = Cambridge University Press | location = Cambridge
 | title = Geometric folding algorithms | title-link=Geometric Folding Algorithms
 | year = 2007}}</ref>

===Huzita–Justin axioms===
{{main|Huzita–Hatori axioms}}

Some [[Compass and straightedge constructions#Impossible constructions|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.<ref>{{cite web |url=http://mars.wne.edu/~thull/omfiles/geoconst.html |title=Origami and Geometric Constructions |author=Tom Hull}}</ref> Paper fold strips can be constructed to solve equations up to degree 4. The Huzita–Justin axioms or Huzita–Hatori axioms are an important contribution to this field of study. These describe what can be constructed using a sequence of creases with at most two point or line alignments at once. Complete methods for solving all equations up to degree 4 by applying methods satisfying these axioms are discussed in detail in ''[[Geometric Origami]]''.<ref name="GO-Geretsch">
{{cite book
|title= Geometric Origami
|title-link= Geometric Origami
|last= Geretschläger
|first= Robert
|year= 2008
|publisher= Arbelos
|location= UK
|isbn= 978-0-9555477-1-3 }}
</ref>

==Constructions==

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.<ref name="GO-Geretsch"/> A regular ''n''-gon can be constructed by paper folding [[if and only if]] ''n'' is a product of distinct [[Pierpont prime]]s, [[power of two|powers of two]], and [[power of three|powers of three]].

===Haga's theorems===
[[File:Haga theorem 1.svg|thumb|200px|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.<ref name=howto>{{cite web
|url=http://origami.gr.jp/Archives/People/CAGE_/divide/02-e.html
|title=How to Divide the Side of Square Paper
|first=Koshiro |last=Hatori |publisher=Japan Origami Academic Society}}</ref><ref name=haga99>K. Haga, Origamics, Part 1, Nippon Hyoron Sha, 1999 (in Japanese)</ref> Surprisingly few folds are necessary to generate large odd fractions. For instance {{Frac|1|5}} can be generated with three folds; first halve a side, then use Haga's theorem twice to produce first {{Frac|2|3}} and then {{Frac|1|5}}.

The accompanying diagram shows Haga's first theorem:

:<math>BQ = \frac{2 AP}{1 + AP}.</math>

The function changing the length ''AP'' to ''QC'' is [[Involution (mathematics)|self inverse]]. Let ''x'' be ''AP'' then a number of other lengths are also rational  functions of ''x''. For example:

{| class="wikitable" border="1" style="text-align:center; width:200px; height:300px" border="1"
|+Haga's first theorem
|-
! AP !! BQ !! QC !! AR !! PQ
|-
| <math>x</math> || <math>\frac{2 x}{1+x}</math> || <math>\frac{1-x}{1+x}</math> || <math>\frac{1-x^2}{2}</math> || <math>\frac{1+x^2}{1+x}</math>
|-
| {{Frac|1|2}} || {{Frac|2|3}} || {{Frac|1|3}} || {{Frac|3|8}} || {{Frac|5|6}}
|-
| {{Frac|1|3}} || {{Frac|1|2}} || {{Frac|1|2}} || {{Frac|4|9}} || {{Frac|5|6}}
|-
| {{Frac|2|3}} || {{Frac|4|5}} || {{Frac|1|5}} || {{Frac|5|18}} || {{Frac|13|15}}
|-
| {{Frac|1|5}} || {{Frac|1|3}} || {{Frac|2|3}} || {{Frac|12|25}} || {{Frac|13|15}}
|}

===A generalization of Haga's theorems===
Haga's theorems are generalized as follows: 
:<math>\frac{BQ}{CQ} = \frac{2 AP}{BP}.</math>
Therefore, BQ:CQ=k:1 implies AP:BP=k:2 for a positive real number k.<ref name=okumura14>{{cite journal|author=Hiroshi Okumura|year=2014|title=A Note on Haga's theorems in paper folding|url=http://forumgeom.fau.edu/FG2014volume14/FG201423.pdf|journal=[[Forum Geometricorum]]|volume=14|pages=241–242}}</ref>

===Doubling the cube===
[[File:Delian origami.svg|thumb|200px|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:<ref name="lang2008">{{cite web |url=http://static.usenix.org/event/usenix08/tech/slides/lang.pdf |title=From Flapping Birds to Space Telescopes: The Modern Science of Origami  |last=Lang |first=Robert J |year=2008 |publisher=Usenix Conference, Boston, MA}}</ref>  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.<ref name=messer86>{{cite journal|author=Peter Messer|year=1986|title=Problem 1054|url=https://cms.math.ca/crux/backfile/Crux_v12n10_Dec.pdf|journal=[[Crux Mathematicorum]]|volume=12|issue=10|pages=284–285|via=Canadian Mathematical Society}}</ref>

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.
{{clear}}

===Trisecting an angle===
[[File:Origami Trisection of an angle.svg|thumb|200px|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.<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211211/SL2lYcggGpc Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20141212181830/https://www.youtube.com/watch?v=SL2lYcggGpc&gl=US&hl=en Wayback Machine]{{cbignore}}: {{Cite web|last=Dancso|first=Zsuzsanna|date=December 12, 2014|title=Numberphile: How to Trisect an Angle with Origami|url=https://www.youtube.com/watch?v=SL2lYcggGpc|access-date=October 2, 2021|website=[[YouTube]]}}{{cbignore}}</ref> This construction, which was reported in 1980, is due to Hisashi Abe.<ref name="lang2008" /><ref name=hullcomp/> The angle CAB is trisected by making two folds: PP', parallel to the base, and 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 [[Congruence (geometry)#Congruence of triangles|congruent]] triangles. Aligning the two points on the two lines is another neusis construction as in the solution to doubling the cube.<ref>{{cite book |title=Origami 5 |chapter=Hands-on Geometry with Origami |author1= Michael J Winckler |author2=Kathrin D Wold |author3=Hans Georg Bock |page=225 |isbn=978-1-56881-714-9 |year=2011 |publisher=CRC Press}}</ref><ref name="hullcomp">{{cite web |last1=Hull |first1=Tom |title=a comparison between straight edge and compass constructions and origami |url=http://origametry.net/omfiles/geoconst.html |website=origametry.net |date=1997}}</ref>
{{clear}}

==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.

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 surface]]s that are not flat.<ref>{{Cite web |url=http://www.siggraph.org/s2008/attendees/design/22.php |title=Siggraph: "Curved Origami" |access-date=2008-10-08 |archive-url=https://web.archive.org/web/20170508164339/http://www.siggraph.org/s2008/attendees/design/22.php |archive-date=2017-05-08 |url-status=dead }}</ref> [[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.

{{anchor|Britney Gallivan}}
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 <math>L=\tfrac{\pi t}{6} (2^n + 4)(2^n - 1)</math>, 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.<ref name=korpal15>{{cite journal|last1=Korpal|first1=Gaurish|title=Folding Paper in Half|journal=At Right Angles |publisher=Teachers of India |date=25 November 2015|volume=4|issue=3|pages=20–23|url=http://teachersofindia.org/en/ebook/folding-paper-half}}</ref>  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 {{convert|4000|ft|m|adj=mid|-long}} 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.<ref name="Mathworld">{{MathWorld | title = Folding | urlname = Folding}}</ref><ref name="d'Agostino">{{Cite book|last=D'Agostino|first=Susan|url=https://books.google.com/books?id=JCbTDwAAQBAJ|title=How to Free Your Inner Mathematician|publisher=[[Oxford University Press]]|year=2020|isbn=9780198843597|pages=22}}</ref>

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.<ref name=hull02>{{Cite journal|jstor = 25678354|title = In search of a practical map fold|last = Hull|first = Thomas|date = 2002|journal = Math Horizons|doi = 10.1080/10724117.2002.11975147|pages = 22–24|volume = 9|issue = 3|s2cid = 126397750}}</ref>

== Computational origami ==
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.<ref name=":2">{{Cite magazine |date=2007-02-12 |title=The Origami Lab |url=http://www.newyorker.com/magazine/2007/02/19/the-origami-lab |access-date=2022-05-09 |magazine=The New Yorker |language=en-US}}</ref> [[Robert J. Lang|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.<ref name=":2" />

=== Research ===
Paper-folding problems are classified as either origami design or origami foldability problems. There are predominantly three current categories of computational origami research: universality results, [[Algorithmic efficiency|efficient decision algorithms]], and [[Computational complexity theory|computational intractability results]].<ref name=":1">Demaine, Erik (2001). "Recent Results in Computational Origami" (PDF).</ref> A universality result defines the bounds of possibility given a particular model of folding. For example, a large enough piece of paper can be folded into any tree-shaped origami base, polygonal silhouette, and polyhedral surface.<ref>{{Cite web |last=Lang |first=Robert |title=A Computational Algorithm for Origami Design |url=https://courses.csail.mit.edu/6.885/fall04/erik_notes/anydpi/L18_paper.pdf}}</ref> When universality results are not attainable, efficient decision algorithms can be used to test whether an object is foldable in polynomial time.<ref name=":1" /> Certain paper-folding problems do not have efficient algorithms. Computational intractability results show that there are no such polynomial-time algorithms that currently exist to solve certain folding problems. For example, it is NP-hard to evaluate whether a given crease pattern folds into any flat origami.<ref>Schneider, Jonathan (December 10, 2004). "Flat-Foldability of Origami Crease Patterns" (PDF).</ref>

In 2017, Erik Demaine of the Massachusetts Institute of Technology and Tomohiro Tachi of the University of Tokyo published a new universal algorithm that generates practical paper-folding patterns to produce any 3-D structure. The new algorithm built upon work that they presented in their paper in 1999 that first introduced a universal algorithm for folding origami shapes that guarantees a minimum number of seams. The algorithm will be included in Origamizer, a free software for generating origami crease patterns that was first released by Tachi in 2008.<ref>{{Cite web |title=Origami anything |url=https://news.mit.edu/2017/algorithm-origami-patterns-any-3-D-structure-0622 |access-date=2022-05-08 |website=MIT News {{!}} Massachusetts Institute of Technology |date=22 June 2017 |language=en}}</ref>

=== Software & tools ===
[[File:Origami kabuto.gif|thumb|Animation of folds to make a Samurai helmet, also called a kabuto. (On a laptop computer, Julia and GLMakie generated<ref>{{cite web |title=Julia and Projective Geometric Algebra |date=31 March 2024 |url=https://medium.com/@olarth/julia-and-projective-geometric-algebra-e2d5abb11519 |location=Julia code animating kabuto is in example 3.4}}</ref> 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 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.<ref>{{Cite web |last=TASON |title=Computational Origami |url=https://langorigami.com/article/computational-origami/ |access-date=2022-05-08 |language=en}}</ref>

There are other software solutions associated with building computational origami models using non-paper materials such as Cadnano in [[DNA origami]].<ref>{{Cite web |title=Cadnano |url=http://cadnano.org/welcome |access-date=2022-05-08 |website=cadnano |language=en}}</ref>

=== Applications ===
Computational origami has contributed to applications in robotics, engineering, biotechnology & medicine, industrial design.<ref>{{Cite web |last1=Magazine |first1=Smithsonian |last2=Morrison |first2=Jim |title=How Origami Is Revolutionizing Industrial Design |url=https://www.smithsonianmag.com/innovation/theres-origami-revolution-industrial-design-180972019/ |access-date=2022-05-08 |website=Smithsonian Magazine |language=en}}</ref> Applications for origami have also been developed in the study of programming languages and programming paradigms, particular in the setting of functional programming.<ref>{{Cite web |last=Gibbons |first=Jeremy |date=2003 |title=Origami Programming |url=https://www.cs.ox.ac.uk/jeremy.gibbons/publications/origami.pdf}}</ref>

Robert Lang participated in a project with researchers at EASi Engineering in Germany to develop automotive airbag folding designs.<ref>{{Cite web |last=TASON |title=Airbag Folding |url=https://langorigami.com/article/airbag-folding/ |access-date=2022-05-08 |language=en}}</ref> In the mid-2000s, Lang worked with researchers at the [[Lawrence Livermore National Laboratory]] to develop a solution for the [[James Webb Space Telescope]], particularly its large mirrors, to fit into a rocket using principles and algorithms from computational origami.<ref>{{Cite web |title=Webb and Origami - Webb Telescope/NASA |url=https://webb.nasa.gov/content/features/origami.html |access-date=2022-05-08 |website=webb.nasa.gov |language=en}}</ref>

In 2014, researchers at the Massachusetts Institute of Technology, Harvard University, and the Wyss Institute for Biologically Inspired Engineering published a method for building self-folding machines and credited advances in computational origami for the project's success. Their origami-inspired robot was reported to fold itself in 4 minutes and walk away without human intervention, which demonstrated the potential for autonomous self-controlled assembly in robotics.<ref>{{Cite journal |last1=Felton |first1=S. |last2=Tolley |first2=M. |last3=Demaine |first3=E. |last4=Rus |first4=D. |last5=Wood |first5=R. |date=2014-08-08 |title=A method for building self-folding machines |url=https://www.science.org/doi/10.1126/science.1252610 |journal=Science |language=en |volume=345 |issue=6197 |pages=644–646 |doi=10.1126/science.1252610 |pmid=25104380 |bibcode=2014Sci...345..644F |s2cid=18415193 |issn=0036-8075}}</ref>

Other applications include [[DNA origami]] and [[RNA origami]], folding of manufacturing instruments, and surgery by tiny origami robots.<ref>{{Cite web |last=Brewin |first=Bob |date=2004-05-10 |title=Computational Origami |url=https://www.computerworld.com/article/2564259/computational-origami.html |access-date=2022-05-08 |website=Computerworld |language=en}}</ref>

Applications of computational origami have been featured by various production companies and commercials. Lang famously worked with Toyota Avalon to feature an animated origami sequence, Mitsubishi Endeavor to create a world entirely out of origami figures, and McDonald's to form numerous origami figures from cheeseburger wrappers.<ref>{{Cite web |title=The Origami Resolution |url=https://www.damninteresting.com/the-origami-resolution/ |access-date=2022-05-08 |website=Damn Interesting |language=en-US}}</ref>

==See also==
* [[Flexagon]]
* [[Lill's method]]
* [[Napkin folding problem]]
* [[Map folding]]
* [[Regular paperfolding sequence]] (for example, the [[dragon curve]])

== Notes and references ==
{{Reflist}}

==Further reading==
* [[Erik Demaine|Demaine, Erik D.]], [http://erikdemaine.org/papers/dthesis/ "Folding and Unfolding"], PhD thesis, Department of Computer Science, University of Waterloo, 2001.
* {{cite book|first= Michael|last=Friedman|publisher=Birkhäuser|year=2018|title=A History of Folding in Mathematics: Mathematizing the Margins|title-link=A History of Folding in Mathematics|series=Science Networks. Historical Studies|volume=59|isbn=978-3-319-72486-7|doi=10.1007/978-3-319-72487-4}}
* {{cite journal |doi=10.2307/2690924|jstor=2690924|last1=Geretschlager|first1=Robert|title=Euclidean Constructions and the Geometry of Origami|journal=Mathematics Magazine|year=1995|volume=68|issue=5|pages=357–371}}
* {{Cite book |last=Haga|first=Kazuo|place=University of Tsukuba, Japan|
title=Origamics: Mathematical Explorations Through Paper Folding|
editor-last=Fonacier|editor-first=Josefina C|
editor2-last=Isoda|editor2-first=Masami|year=2008|
isbn=978-981-283-490-4|publisher=World Scientific Publishing }}
* {{cite book|author=Lang, Robert J.|author-link=Robert J. Lang|title=Origami Design Secrets: Mathematical Methods for an Ancient Art|publisher=A K Peters| year=2003| isbn=978-1-56881-194-9}}
* [[David Dureisseix|Dureisseix, David]], [https://hal.archives-ouvertes.fr/hal-00321386 "Folding optimal polygons from squares"], ''Mathematics Magazine'' 79(4): 272–280, 2006. {{doi|10.2307/27642951}}
* [[David Dureisseix|Dureisseix, David]], [https://hal.archives-ouvertes.fr/hal-00687311 "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}}

==External links==
{{commons category|Origami mathematics}}
* {{Cite web|url=http://mars.wne.edu/~th297133/origamimath.html|
title=Origami Mathematics Page|
author=Dr. Tom Hull|author-link=Tom Hull (mathematician) }}
* [http://www.cut-the-knot.org/pythagoras/PaperFolding/index.shtml Paper Folding Geometry] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/pythagoras/PaperFolding/SegmentDivision.shtml Dividing a Segment into Equal Parts by Paper Folding] at [[cut-the-knot]]
* [https://web.archive.org/web/20051102085038/http://pomonahistorical.org/12times.htm Britney Gallivan has solved the Paper Folding Problem]
* [http://plus.maths.org/content/power-origami Overview of Origami Axioms]
* [http://www.mariocigada.com/mariodocs/origanova_e.html Introduction to Statistics with Origami] by [http://www.mariocigada.com/ Mario Cigada]

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