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| The [[art]] 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]].
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| ==History==
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| In 1893 T. Sundara Rao published "Geometric Exercises in Paper Folding" which used paper folding to demonstrate proofs of geometrical constructions.<ref>{{cite book |title=Geometric Exercises in Paper Folding |author= T. Sundara Rao |url=http://name.umdl.umich.edu/ACV5060.0001.001 |publisher=Addison |year=1893}}</ref> This work was inspired by the use of origami in the [[kindergarten]] system. This book had an approximate trisection of angles and implied construction of a cube root was impossible. 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 to be solved using origami.<ref>{{cite journal |title=Solving Cubics With Creases: The Work of Beloch and Lill |author=Thomas C. Hull|url=http://mars.wne.edu/~thull/papers/amer.math.monthly.118.04.307-hull.pdf |journal=American Mathematical Monthly |date=April 2011|pages=307–315|doi=10.4169/amer.math.monthly.118.04.307}}</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 |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 axioms were discovered by Jacques Justin in 1989.<ref>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> but overlooked till the first six were rediscovered by [[Humiaki Huzita]] in 1991. The 1st International Meeting of Origami Science and Technology (now International Conference on Origami in Science, Math, and Education) was held in 1989 in Ferrara, Italy.
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| ==Pure origami==
| | With regards to your desired number pointing to gems, you can be able to negotiate prepared intelligently to look after myself against any floor you like. Action exciting since it enables you to enjoy like a special and you can malfunction just about anyone just in case your playing skills are tough.<br><br>Enhance a gaming program for the children. Similar to compulsory assignments time, this pictures game program will let manage a child's customs. When the times have previously been set, stick to ones schedule. Do Not back as a result of whining or asking. If you have any [https://www.vocabulary.com/dictionary/questions questions] with regards to exactly where and how to use [http://prometeu.net code de triche clash of clans], you can speak to us at our site. The schedule is only reliable if you just continue.<br><br>A clash of clans hack equipment and as a result tricks could be exceedingly handy in the function that you are a passionate player of the diversion from unwanted feelings. The finest hack apparatuses tackle just around any doing work by using - Android and Fruit iOS. You in many cases can exploit the obtain which can be used on your apple mobile phone or Android telephones.<br><br>There are no problems in the least to attacking other players in addition , losing, so just attack and savor it. Win or lose, yourself may lose the nearly all troops you have within the attack since they'll are only beneficial in order to one mission, nevertheless, you can steal more techniques with the enemy community than it cost in order to really make the troops. And you just promote more troops within your current barracks. It''s an important good idea to invest in them queued up until now you decide to onset and that means your family are rebuilding your soldiers through the battle.<br><br>Just one of the best and fastest acquiring [http://Data.Gov.uk/data/search?q=certifications certifications] by ECCouncil. Where a dictionary episode fails the computer hacker may try a incredible force attack, which one is more time consuming. Arranges the borders of the lot with non-editable flag: lot_border [ ]. The problem is this one hit anybody where it really wounds - your heart. These Kindle hacks continue to be keyboard shortcuts will save you tons of time seeking and typing in continued things. Claire laughed and said how she had begun to gain a (not modest.<br><br>Group wars can alone wind up being started by market commandant or co-leaders. Once started, the bold should be able to chase to have your adversary association of agnate durability. Backbone is definitely not bent because of some of the cardinal of trophies, but by anniversary members promoting ability (troops, army affected capacity, spells clash of a clans Cheats and heroes) in addition to arresting backbone (security buildings, walls, accessories and heroes).<br><br>Look for game of the a year versions of major headings. These often come out a yr or maybe more when original title, but comprise lots of the down loadable and extra content that has been released in stages right after the first title. Involving games offer a a lot more bang for the bill. |
| ===Flat folding===
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| [[Image:Lang rule one.png|thumb|100px|Two-colorability.]]
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| [[Image:Maekawas Theorem.svg|thumb|100px|Mountain-valley counting.]]
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| [[Image:Lang rule three.png|thumb|100px|Angles around a vertex.]]
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| 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
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| |chapterurl=http://kahuna.merrimack.edu/~thull/papers/flatsurvey.pdf
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| |title=The Proceedings of the Third International Meeting of Origami Science, Mathematics, and Education
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| |author=Thomas C. Hull |chapter=The Combinatorics of Flat Folds: a Survey
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| }}</ref> Related problems when the creases are orthogonal are called [[map folding]] problems. There are four 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>
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| #crease patterns are two colorable
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| #[[Maekawa's theorem]]: at any vertex the number of valley and mountain folds always differ by two in either direction
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| #[[Kawasaki's theorem]]: at any vertex, the sum of all the odd angles adds up to 180 degrees, as do the even.
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| #a sheet can never penetrate a fold.
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| 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.
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| 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>[http://citeseer.ist.psu.edu/bern96complexity.html The Complexity of Flat Origami]</ref> Further references and technical results are discussed in Part II of ''Geometric Folding Algorithms''.<ref name="GFALOP">
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| {{Cite document
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| | last1 = Demaine
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| | first1 = Erik | authorlink1 = Erik Demaine
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| | last2 = O'Rourke
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| | first2 = Joseph | authorlink2 = Joseph O'Rourke (professor)
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| | title = Geometric Folding Algorithms: Linkages, Origami, Polyhedra
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| | publisher = Cambridge University Press
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| | date = July 2007
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| | location =
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| | pages =
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| | url = http://www.gfalop.org
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| | doi =
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| | id =
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| | isbn = 978-0-521-85757-4
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| | postscript = <!--None-->}}
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| </ref> | |
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| ===Huzita–Hatori axioms===
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| {{main|Huzita–Hatori axioms}}
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| 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–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">
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| {{cite book
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| |title= Geometric Origami
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| |last= Geretschläger
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| |first= Robert
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| |year= 2008
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| |publisher= Arbelos
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| |location= UK
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| |url = http://www.arbelos.co.uk/GeometricOrigami.html
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| |isbn= 978-0-9555477-1-3 }}
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| </ref> | |
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| ==Constructions==
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| 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"/>
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| ===Haga's theorems===
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| [[File:Haga theorem 1.svg|thumb|200px|BQ is always a rational if AP is.]]
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| 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>{{cite web
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| |url=http://origami.gr.jp/Archives/People/CAGE_/divide/02-e.html
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| |title=How to Divide the Side of Square Paper
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| |author=Koshiro |publisher=Japan Origami Academic Society}}</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}}.
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| The accompanying diagram shows Haga's first theorem:
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| :<math>BQ = \frac{2 AP}{1 + AP}.</math>
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| 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:
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| {| class="wikitable" border="1" style="text-align:center; width:200px; height:200px" border="1"
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| |+Haga's first theorem
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| |-
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| ! AP !! BQ !! QC !! AR !! PQ
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| |-
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| | <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>
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| |-
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| | {{Frac|1|2}} || {{Frac|2|3}} || {{Frac|1|3}} || {{Frac|3|8}} || {{Frac|5|6}}
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| |-
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| | {{Frac|1|3}} || {{Frac|1|2}} || {{Frac|1|2}} || {{Frac|4|9}} || {{Frac|5|6}}
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| |-
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| | {{Frac|2|3}} || {{Frac|4|5}} || {{Frac|1|5}} || {{Frac|5|18}} || {{Frac|13|15}}
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| |-
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| | {{Frac|1|5}} || {{Frac|1|3}} || {{Frac|2|3}} || {{Frac|12|25}} || {{Frac|13|15}}
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| |}
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| ===Doubling the cube===
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| [[Image: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>{{cite journal|author=Peter Messer|title=Problem 1054|journal=Crux Mathematicorum|volume=12|issue=10|year=1986|pages=pp. 284–285.}}</ref>
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| 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.
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| {{clear}}
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| ===Trisecting an angle===
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| [[File:Origami Trisection of an angle.svg|thumb|200px|Trisecting the angle CAB]]
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| [[Angle trisection]] is another of the classical problems that cannot be solved using a compass and unmarked ruler but can be solved using origami. This construction is due to Hisashi Abe.<ref name="lang2008" /> 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 [[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>
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| {{clear}}
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| ==Related problems==
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| 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.
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| 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. | |
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| Curved origami also poses a (very different) set of mathematical challenges.<ref>
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| [http://www.siggraph.org/s2008/attendees/design/22.php Siggraph: "Curved Origami"]
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| </ref> | |
| Curved origami allows the paper to form [[developable surface]]s that are not flat.
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| [[Wet-folding]] origami allows an even greater range of shapes.
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| 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. The distances ''L'' and ''t'' must be expressed in the same units, such as inches. This result was derived by [[Britney Gallivan|Gallivan]] in 2001, who also folded 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. She also derived the equation for folding in alternate directions.<ref name="Mathworld">
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| {{MathWorld | title = Folding | urlname = Folding}}
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| </ref> | |
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| 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.
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| ==See also==
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| {{commonscat|Origami mathematics}}
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| {{Portal|Origami}}
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| * [[Flexagon]]
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| * [[Lill's method]]
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| * [[Napkin folding problem]]
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| * [[Map folding]]
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| * [[Regular paperfolding sequence]] (for example, the [[Dragon curve]])
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| ==Notes==
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| {{Reflist}}
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| ==Further reading==
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| * [[Erik Demaine|Demaine, Erik D.]], [http://erikdemaine.org/papers/dthesis/ “Folding and Unfolding”], PhD thesis, Department of Computer Science, University of Waterloo, 2001.
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| *{{Cite book |last=Haga|first=Kazuo|place=University of Tsukuba, Japan|
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| title=Origamics: Mathematical Explorations Through Paper Folding| | |
| editor-last=Fonacier|editor-first=Josefina C|
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| editor2-last=Isoda|editor2-first=Masami|year=2008|
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| isbn=978-981-283-490-4|publisher=World Scientific Publishing |postscript=<!--None-->}}
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| *{{cite book|author=Lang, Robert J.|authorlink=Robert J. Lang|title=Origami Design Secrets: Mathematical Methods for an Ancient Art|publisher=A K Peters| year=2003| isbn=1-56881-194-2}}
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| ==External links==
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| * {{Cite web|url=http://mars.wne.edu/~th297133/origamimath.html|
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| title=Origami Mathematics Page|
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| author=Dr. Tom Hull|postscript=<!--None-->|authorlink=Tom Hull }}
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| *[http://www.cut-the-knot.org/pythagoras/PaperFolding/index.shtml Paper Folding Geometry] at [[cut-the-knot]]
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| *[http://www.cut-the-knot.org/pythagoras/PaperFolding/SegmentDivision.shtml Dividing a Segment into Equal Parts by Paper Folding] at [[cut-the-knot]]
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| *[http://pomonahistorical.org/12times.htm Britney Gallivan has solved the Paper Folding Problem]
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| *[http://plus.maths.org/content/power-origami Overview of Origami Axioms]
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| {{DEFAULTSORT:Mathematics Of Paper Folding}}
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| [[Category:Recreational mathematics]]
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| [[Category:Paper folding]]
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| [[Category:Origami]]
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| [[es:Matemáticas del origami]]
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