Оригами

Оригами модели

Савијање оригами модела

A group of Japanese schoolchildren dedicate their contribution of Thousand origami cranes at the Sadako Sasaki memorial in Hiroshima.

Оригами (јап. 折り紙; ori = савијање + kami = папир) је традиционална јапанска вештина креирања модела од папира. Традицинално се користи квадрат али постоји велики број модела који се прави и од другачијих облика папира - правоугаоника, троугла, осмоугаоника итд.

The small number of basic origami folds can be combined in a variety of ways to make intricate designs. The best-known origami model is the Japanese paper crane. In general, these designs begin with a square sheet of paper whose sides may be of different colors, prints, or patterns. Traditional Japanese origami, which has been practiced since the Edo period (1603–1867), has often been less strict about these conventions, sometimes cutting the paper or using nonsquare shapes to start with. The principles of origami are also used in stents, packaging, and other engineering applications.

Историја

Не постоје прецизни подаци о томе када је оригами настао. Најчешће се везује за изум папира у Кини негде око 2. века н. е. Иако је тамо највероватније и настао, оригами је прави процват доживео у Јапану, где се и третира као национална уметност. Поред Јапана, ова вештина се појавила и у другим деловима света, на пример, у Шпанији где је позната под именом Papiroflexia.

Већ у 8. веку, оригами је постао саставни део разних церемонија у Јапану. Самураји су размењивали поклоне који су на себи имали украсе „ноши“ - савијене траке папира. За време обреда шинтоистичких венчања, коришћени су оригами лептири који су симболизовали младенце.

In 1893, Indian civil servant T. Sundara Rao 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. Rao demonstrated an approximate trisection of angles and implied construction of a cube root was impossible.

The Beloch fold

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

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

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

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

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

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. In 1985 Miura reported a method of packaging and deployment of large membranes in outer space, and as late as 2012 this technique had become standard operating procedure for orbital vehicles.

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.

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. 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.

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

Mountain-valley counting

In 1996, Marshall Bern and Barry Hayes showed to be an NP-complete problem the assignation of a crease pattern of mountain and valley folds in order to produce a flat origami structure starting from a flat sheet of paper.

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

In 2001, among other more mathematical results Britney Gallivan folded first a bedsheet then a sheet of gold foil in half 12 times, contrary to the belief that paper of any size could be folded at most eight times.

In 2002, Belcastro and Hull brought to the theoretical origami the language of affine transformations, with an extension from $R$² to $R$³ in only the case of single-vertex construction.

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

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

Материјал

Оригами се у принципу прави од папира, иако се могу користити и другачији материјали (тканина и сл.)

За вежбу и неке једноставне моделе често се користи папир за фотокопирање стандардне грамаже 70–90 g/m². Поред тога, могуће је користити и разне друге врсте папира - фолију, папир за увијање, хамер и тд. Постоји и специјализован папир за оригами који је најчешће двобојан и већ исечен у облик квадрата.

У Јапану се често користи „ваши“ - специјалан папир чвршће структуре направљен од пулпе добијене из коре неколико карактеристичних дрвенастих врста које расту у Јапану.

Постоји и посебна грана оригамија која користи новчанице за прављење модела и то најчешће амерички долар.

Види још

Киригами

Литература

• Kunihiko Kasahara (1988). Origami Omnibus: Paper Folding for Everybody. Tokyo: Japan Publications, Inc. ISBN 4-8170-9001-4
• Kunihiko Kasahara and Toshie Takahama (1987). Origami for the Connoisseur. Tokyo: Japan Publications, Inc. ISBN 0-87040-670-1
• Satoshi Kamiya (2005). Works by Satoshi Kamiya, 1995–2003. Tokyo: Origami House
• Kunihiko Kasahara (2001). Extreme Origami. ISBN 0-8069-8853-3
• Michael LaFosse. Origamido : Masterworks of Paper Folding ISBN 978-1564966391
• Nick Robinson (2004). Encyclopedia of Origami. Quarto. ISBN 1-84448-025-9. A book full of stimulating designs.
• Demaine, Erik D., "Folding and Unfolding", PhD thesis, Department of Computer Science, University of Waterloo, 2001.
• Friedman, Michael (2018). A History of Folding in Mathematics: Mathematizing the Margins. Science Networks. Historical Studies. 59. Birkhäuser. ISBN 978-3-319-72486-7. doi:10.1007/978-3-319-72487-4. 
• Geretschlager, Robert (1995). „Euclidean Constructions and the Geometry of Origami”. Mathematics Magazine. 68 (5): 357—371. JSTOR 2690924. doi:10.2307/2690924. 
• Haga, Kazuo (2008). Fonacier, Josefina C; Isoda, Masami, ур. Origamics: Mathematical Explorations Through Paper Folding. University of Tsukuba, Japan: World Scientific Publishing. ISBN 978-981-283-490-4. 
• Lang, Robert J. (2003). Origami Design Secrets: Mathematical Methods for an Ancient Art. A K Peters. ISBN 978-1-56881-194-9. 
• Dureisseix, David, "Folding optimal polygons from squares", Mathematics Magazine 79(4): 272–280, 2006. doi:10.2307/27642951
• Dureisseix, David, "An Overview of Mechanisms and Patterns with Origami", International Journal of Space Structures 27(1): 1–14, 2012. doi:10.1260/0266-3511.27.1.1

Спољашње везе

• Origami Resource Center
• Origami Instructions
• Classical origami on Moscow origami club portal (рус)
• OrigamiWiki (рус.)
• GiladOrigami.com, contains many book reviews
• WikiHow on how to make origami
• Origami USA, many resources, especially for folders in the USA
• British Origami Society, many resources, especially for folders in the UK
• Between the Folds, documentary film about origami and origami artists
• Lang, Robert (фебруар 2008). „The math and magic of origami” (video). TED ED. Приступљено 6. 4. 2013. CS1 одржавање: Формат датума (веза)
• Robert Lang (16. 3. 2019). „See a NASA Physicist's Incredible Origami” (video). Southwest Daily News (на језику: енглески). CS1 одржавање: Формат датума (веза)
• Engineering with Origami, YouTube video by Veritasium about uses of origami for structural engineering