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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]].
| | BSOD or the Blue Screen of Death, (sometimes known as blue screen bodily memory dump), is an error which occurs on a Windows program - whenever the computer really shuts down or automatically reboots. This error can happen merely as your computer is booting up or certain Windows application is running. When the Windows OS discovers an unrecoverable error it hangs the system or leads to memory dumps.<br><br>We should recognize some simple plus cheap methods which can resolve the issue of your computer plus speed it up. The earlier you fix it, the less damage the computer gets. I usually tell about certain valuable techniques that will enable we to accelerate you computer.<br><br>So what could you look for when we compare registry products. Many of the registry cleaners available now, have very similar qualities. The key ones which we should be hunting for are these.<br><br>Review a files plus clean it up regularly. Destroy all unwanted plus unused files because they only jam your computer program. It can surely better the speed of your computer plus be thoughtful which a computer never afflicted by a virus. Remember always to update the antivirus software every time. If you never use your computer surprisingly often, you are able to take a free antivirus.<br><br>Another choice when arresting the 1328 error is to wash out your PC's registry. The registry is important because it is actually where settings plus files employed by Windows for running are stored. As it is frequently chosen, breakdowns plus cases of files getting corrupted are not uncommon. Additionally as a result of the technique it is configured, the "registry" gets saved in the wrong fashion frequently, that makes the system run slow, eventually causing your PC to suffer from a series of errors. The best way one can use in cleaning out registries is to utilize a reliable [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities] system. A registry cleaner may find out and repair corrupted registry files and settings allowing one's computer to run normally again.<br><br>The program is designed and built for the purpose of helping we accomplish tasks and not be pestered by windows XP error messages. When there are errors, what do we do? Some persons pull their hair plus cry, whilst those sane ones have their PC repaired, while those certainly wise ones research to have the mistakes fixed themselves. No, these errors were not equally tailored to rob you off the income plus time. There are points to do to actually avoid this from happening.<br><br>Most likely should you are experiencing a slow computer it could be a couple years old. You furthermore may not have been told which whilst you employ a computer everyday; there are certain elements which it needs to continue running in its ideal performance. We additionally could not even own any diagnostic tools that may receive the PC running like new again. So do not let that stop you from getting a system cleaned. With access to the internet you are able to find the tools that will help you get a program running like unique again.<br><br>So in summary, whenever comparing registry cleaning, make sure the one we choose offers you the following.A backup and restore center, quickly procedure, automatic deletion center, start-up administration, an convenient means of contact and a funds back guarantee. |
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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==
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| ===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]]
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| 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. | |
| {{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>
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| 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|
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| 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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BSOD or the Blue Screen of Death, (sometimes known as blue screen bodily memory dump), is an error which occurs on a Windows program - whenever the computer really shuts down or automatically reboots. This error can happen merely as your computer is booting up or certain Windows application is running. When the Windows OS discovers an unrecoverable error it hangs the system or leads to memory dumps.
We should recognize some simple plus cheap methods which can resolve the issue of your computer plus speed it up. The earlier you fix it, the less damage the computer gets. I usually tell about certain valuable techniques that will enable we to accelerate you computer.
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Review a files plus clean it up regularly. Destroy all unwanted plus unused files because they only jam your computer program. It can surely better the speed of your computer plus be thoughtful which a computer never afflicted by a virus. Remember always to update the antivirus software every time. If you never use your computer surprisingly often, you are able to take a free antivirus.
Another choice when arresting the 1328 error is to wash out your PC's registry. The registry is important because it is actually where settings plus files employed by Windows for running are stored. As it is frequently chosen, breakdowns plus cases of files getting corrupted are not uncommon. Additionally as a result of the technique it is configured, the "registry" gets saved in the wrong fashion frequently, that makes the system run slow, eventually causing your PC to suffer from a series of errors. The best way one can use in cleaning out registries is to utilize a reliable tuneup utilities system. A registry cleaner may find out and repair corrupted registry files and settings allowing one's computer to run normally again.
The program is designed and built for the purpose of helping we accomplish tasks and not be pestered by windows XP error messages. When there are errors, what do we do? Some persons pull their hair plus cry, whilst those sane ones have their PC repaired, while those certainly wise ones research to have the mistakes fixed themselves. No, these errors were not equally tailored to rob you off the income plus time. There are points to do to actually avoid this from happening.
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