A. Introduction
Origami is the art of paper-folding into works of sculpture preferably without the aid of scissors or glue. The methods of folding paper into pretty forms and images made origami a highly creative art. Nowadays with the modern advancements, origami reveals a rich geometric structure that gives us the infinite possibilities of exploring paper-folding. Traditional origami is done without cutting or pasting. However, in the recent creative origami, scissors and even paste are used to help form more complicated designs. At a lower level, origami is just regarded as a hobby that can bring enjoyment. At a higher level, it actually involves a series of mathematical concepts like origami geometry and symmetry. From the animals to polygons, origami displays an immeasurable creation of art. The beauty of origami lies in the result of a totally attractive piece of mathematical artwork which is created from a simple, flat sheet of square paper by using almost entirely folds of corners, creases, and edges. B. History of Origami 1. Early History of Origami:
The name “origami” is originated from Japanese word, “ori” which means folding and “kami” which means paper. When combined together, it formed the word origami. Some say origami originated in China around 2000 years ago. But it is probably wrong. This opinion is based on the conjecture that origami started right after the invention of paper, for which they have no evidence. The paper of Former Han dynasty shows no trace of origami.
The Chinese character for paper, zhi, originally stood for writing material made of silk. The origin of the Japanese word for paper, kami, is said to be birch tree, kaba, or strips of wood or bamboo, kan. Both of them are also writing material. These suggest that paper was primarily writing material, not folding. Others say origami originated from Japan in Heian era. Again, it is probably wrong. They refer to a story of Abe-no Seimei who made a paper bird and turned it to a real one, or another story about Fujiwara-no Kiyosuke who sent his ex-girlfriend a fake frog. There is no reason, however, for believing that they folded paper to make them. In Japan, they use wrapping paper called tatogami or tato. Today they mainly wrap kimono with it. It actually dates back to Heian era. But it is by no means an example of origami, since it is folded just squarely. They use paper strips, shide or heisoku, and paper dolls, hitogata, in Shinto. They are also old. However, they were never made of paper in ancient Japan. In addition, they are not necessarily folded even now. They can see no relationship
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between Japanese religion and the origin of origami. The Japanese words for paper and gods have the same spelling, kami. But their pronunciation they are different in old Japanese. They use the word origami from Heian era in Japan. But it originally refers to a form of writing. An origami is a landscape piece of paper folded in half latitudinally. They usually write letters or lists on it. In today's Japan, origami-tsuki (with origami) means authentic because connoisseurs write their appraisal on the origami since Edo era. They did not call paper folding origami in Japan until Showa era. Origami was called "orisue" or "orikata" in Edo era, and "orimono" from the end of Edo era to the early Showa era. In Edo period (1600-1868) new methods were develop to effectively mass produce paper. This is when origami began to fully develop into the art that we know today. Origami instructions were passed down person to person and no diagrams were written until 1797 when a book called Senbazuru Orikata (How to Fold 1000 Cranes) was published. In Japanese mythology the Crane was considered a sacred bird. As origami gained in popularity, the custom that one would receive a wish or receive good luck from folding a thousand Cranes developed.
2. Traditional Origami:
Both Japanese and European had their origami when Japan closed its border. The origami in two regions were fairly independent. The Meiji Restoration and following exchange between Japan and Europe caused fusion of East and West origami. Japanese imported Fröbelian kindergarten movement, which contained European classic origami, when they re-built the educational system after the European one. On the other hand, Western kindergarten adopted Japanese classic origami. Thus Japanese and European classic origami were mixed. The repertoire of origami evolved here has come down until now and formed the core of traditional origami. Japanese also started to produce origami paper, a square of Western paper colored on one side, because of the needs of kindergarten to teach Fröbelian origami. Since Meiji era, new models have been added in traditional origami, and many of them are suitable to fold with origami paper. On the other hand, many models suitable to fold with washi were dismissed. The models of traditional origami travel a long distance in a short time, sometimes beyond borders, as people move. Japanese Orizuru migrated to Europe and became Flapping Bird in the first years of Meiji era. Then Miguel de Unamuno, who
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was active from the end of 19th century to the early 20th, made many models based on Flapping Bird.
In Europe, they did not use the word origami until 1950s. Origami was called "papierfalten" in German and "paper folding" in English. When Japan imported Fröbelian origami, they were translated to "shoshi," "tatamigami," or "kamitatami" at the kindergarten, and to "origami-zaiku" or "origami" at the primary schools. But these words did not spread out of the educational system. In Spanish, "pajarita" means not only origami bird but also origami in general. Traditional origami has been born and brought up in the cultural exchange between East and West. It is not a Japanese original culture but intrinsically a hybrid of Japan and Europe. Although it is most popular in Japan, it has been inherited in Europe, the Americas, China, and so forth since 19th century or the early 20th. 3. Modern Origami:
Modern origami, which started in 20th century, is based on the completely different paradigm. The folding sequences of modern origami are regarded as "models" "designed" by "origami creators." The father of modern origami would be Uchiyama Koko, as he patented his origami models. Today some people believe that origami models should be copyrighted. The idea that particular persons have intellectual property in folding sequences is typical of modern origami.
In modern origami, the creativity is attributed to the designers, and the appreciation to the folders. They prefer the models which have not only good final shapes but also good sequences. In addition, they put importance on reproducibilit y of the model, that is, folders are supposed to make the same shape as intended by the designer.
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The diagrams, which represent the folding sequence of a model, are important in modern origami, as they represent the model itself. They are supposed to show the entire sequence. We have similar kind of representation in Japanese classic origami, but they did not describe the entire sequence. In the modern origami, some emphasize the aspect of origami as the puzzle reproducing the shapes of objects under a certain rule. The most common rule is to fold one sheet of square paper without cutting or gluing. Behind the rule, there is an implicit premise that origami models should be folded with origami paper. They make a point of easiness of origami, that is, we need nothing but origami paper to do origami. Thus, a model made of more than one sheet of paper is regarded as good when it is made of sheets in the same size and can be assembled without glue. In 1950s and 60s, an international origami circle was established by creators and folders such as Yoshizawa Akira, Takahama Toshie, Honda Isao, Robert Harbin, Gershon Legman, Lillian Oppenheimer, Samuel Randlett, Vicente SolórzanoSagredo, and so forth. They have advanced popularization of origami through their community. They published the origami models of the designers from Japan, Europe, and the Americas in Japanese and English. They also founded national and local organizations then "Origami" became an universal word. 4. Mathematics Origami
We often apply the first part of a certain model's sequence and design different origami models. Consequently, many models have some halfway shapes in common. Such halfway shapes are called bases when they are arranged according to the geometrical analysis. Among the first surveys of the bases were that of Uchiyama Koko in 1930s and that of Vicente Solórzano-Sagredo in 1940s. The new models in modern origami depends heavily on a few established bases. They use Bird base, which is the halfway shape of Orizuru, in creating not only birds but also animals or flowers. They hardly invent new bases, although they sometimes make variations such as Bird base folded from a triangle or a combination of Bird base and Frog base. When we fold a base and unfold it, we get a crease pattern. Geometrical study of the crease pattern has been made since 1980s, and it paved the way for the invention of new bases. Now the meaning of the base become completely different. Suppose a creator designs a new model, say Pegasus, s/he would not choose one from existing bases but make Pegasus base. Maekawa Jun and Peter Engel independently started such mathematical origami. Both of them noticed that the crease patterns of established bases consist of particular triangles and rectangles. They divided a crease pattern into these "atoms," and rearrange them to make new crease patterns. In other words, they designed new models before they fold them. The advanced theory has been developed by Meguro Toshiyuki, Kawahata Fumiaki, Robert Lang, and others. In this theory, a base is regarded as a set of independent areas and distinguished by the length and arrangement of the areas. They devised algorithms that generate the crease pattern of the base from an arbitrary length and arrangement of areas. Lang's TreeMaker is a computer
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program which supports origami design based on his algorithm. Combining these design method, we can make complex models with only one sheet of square paper and without scissors. Thus, the aspect of origami as the puzzle is more and more emphasized in mathematical origami. That is, they compete in designing realistic or complex models under the rule of one sheet of square with no cut. In addition, they regards the crease pattern as an important part of the model besides the final shape and the sequence.
5. Historical Figures of Origami:
Akira Yoshizawa was born on March 14, 1911. His parents were dairy farmers, but he moved to Tokyo when he was just 13 years old to take a job in a factory. When he was in his early 20s, he was promoted from his factory worker position to a job as a technical draftsman. As part of his duties, he was responsible for teaching new employees basic geometry. He decided to use origami, which he had learned as a child, as a teaching tool to make these lessons easier to understand. In 1937, Akira Yoshizawa quit his factory job to practice origami on a fulltime basis. He essentially lived in poverty for most of the next two decades, making a meager living by selling tsukudani on a door-to-door basis. During World War II, Akira Yoshizawa served in the army medical corps in Hong Kong. He made origami models to cheer up the sick patients, but eventually fell ill himself and was sent back to Japan. In 1951, a Japanese magazine asked Akira Yoshizawa to fold models of the 12 signs of the Japanese zodiac. This was a turning point in his career, since the exposure led to several exhibitions of his work and the publication of 18 different origami books. In 1954, Akira Yoshizawa founded the International Origami Centre in Tokyo. The Centre helps promote awareness of origami by arranging exhibitions, demonstrations, and instructional classes. In 1956, Akira Yoshizawa married his wife Kiyo. She served as his manager and taught origami alongside him until his death. Akira Yoshizawa's incredible origami skill afforded him many opportunities to travel around the world, which made it possible for him to serve as a goodwill ambassador for the Japanese government. In 1983, he was named to the Order of the Rising Sun. This is one of the highest honors for a citizen of Japan. On March 14,
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2005, Akira Yoshizawa died from complications of pneumonia. This was his 94th birthday.On March 14, 2012, Google honored Akira Yoshizawa on what would have been his 101st birthday with an origami doodle on their homepage. The doodle was created by Dr. Robert J. Lang. Akira Yoshizawa Origami Models and Exhibitions Even though Akira Yoshizawa was a self-taught origami artist, his work was eventually exhibited in shows around the world. His origami appeared in exhibits at the Cooper Union in New York, the Stedelijk Museum in Amsterdam, and the Louvre. Akira Yoshizawa estimated he created 50,000 different origami models over the course of his career. However, only a handful of these were ever diagrammed and published. Simple and elegant lines were the defining features of Akira Yoshizawa's origami models. He never used scissors, glue, or additional embellishments when creating his origami designs. His lumbering gorillas, flying dragons, and graceful cranes were sculptural art. His goal was to convey emotion and feeling – even if this didn't result in an entirely realistic representation of his subject. Invention of the Wet-Folding Technique Although Akira Yoshizawa pioneered many different origami techniques, wetfolding is one of his most significant contributions. This technique involves slightly dampening the paper before making a fold. Wet-folding allows the paper to be manipulated more easily, resulting in finished origami models that have a rounder and more sculpted look. The ability to create origami with a more realistic appearance was an important advancement in paper folding, since it took models away from the realm of simple crafts and towards true artistic expression. Wet-folding is most often used with thicker paper, however. Normal origami paper is very thin and thus prone to tearing when using the wet-folding technique. Creation of the Yoshizawa-Randlett System The Yoshizawa-Randlett system of notations is a standardized way of diagramming the steps involved in folding a particular origami model. In 1954, Akira Yoshizawa's Atarashi Origami Geijutsu (New Origami Art) used a diagramming system that included dotted and dashed lines to indicate mountain and valley folds, plus symbols such as the markings for “inflate” and “round.” This caught the attention of Samuel Randlett and Robert Harbin, who added a few additional symbols to develop the complete notation system that is still used by paper folders around the world today.
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Robert J. Lang (born May 4, 1961) is an American physicist who is also one of the foremost origami artists and theorists in the world. He is known for his complex and elegant designs, most notably of insects and animals. He has studied the mathematics of origami and used computers to study the theories behind origami. He has made great advances in making real-world applications of origami to engineering problems. Lang was introduced to origami at the age of six by a teacher who had exhausted other methods of keeping him entertained in the classroom. By his early teens, he was designing original origami patterns. Lang used origami as an escape from the pressures of undergraduate studies. While studying at Caltech, Lang came into contact with other origami masters such as Michael LaFosse, John Montroll, Joseph Wu, and Paul Jackson through the Origami Center of America, now known as Origami USA. While in Germany for postdoctoral work, Lang and his wife were enamored of Black Forest cuckoo clocks, and he became a sensation in the origami world when he successfully folded one after three months of design and six hours of actual folding. Lang takes full advantage of modern technology in his origami, including using a laser cutter to help score paper for complex folds. Lang is recognized as one of the leading theorists of the mathematics of origami. He has developed ways to algorithmatize the design process for origami, and is the author of the proof of the completeness of the Huzita – Hatori axioms. Lang specializes in finding real-world applications for the various theories of origami he has developed. These included designing folding patterns for a German airbag manufacturer. He has worked with the Lawrence Livermore National Laboratory in Livermore, California, where a team is developing a powerful space telescope, with a 100 m (328 ft) lens in the form of a thin membrane. Lang was engaged by the team to develop a way to fit the tremendous lens, known as the Eyeglass, into a small rocket in such a way that the lens can be unfolded in space and will not suffer from any permanent marks or creases. Lang is the author or co-author of eight books and many articles on origami.
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resources http://mmorganportfolio.weebly.com/uploads/7/8/8/6/7886538/informal_and_formal.pdf http://courses.csail.mit.edu/6.849/fall10/lectures/L20_images.pdf http://math.uttyler.edu/nathan/classes/senior-seminar/JaemaKrier.pdf http://www.origami.gr.jp/Archives/People/CAGE_/divide/01-e.html http://www.takayaiwamoto.com/Greek_Math/Delian/Haga_Delian.html http://origami.about.com/od/History-Of-Origami/a/Akira-Yoshizawa-Origami-Biography.htm http://en.wikipedia.org/wiki/Robert_J._Lang http://origami.ousaan.com/library/historye.html
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