Origami

Famous Paper Folding Art of Japan
Origami is the traditional Japanese art of folding paper to make models of plants, animals, daily-use tools and amongst others. It also refers to a folded paper model itself or a piece of square paper used exclusively for origami. Recently, it has been recognized from its artistic aspect, and original, elaborate, designs of beauty have been created. Due to the geometric feature of origami, it has been studied as a field of mathematics.

The Universe of One Sheet of Paper
In the past, colorful paper called chiyogami was used for origami. Therefore, origami paper can be called chiyogami. Recently, chiyogami has been sold as traditional art work. In many cases, origami models are made of a sheet of paper without using scissors or glue, but scissors are used to cut paper for certain designs which require two sheets of paper, for example shuriken (a ninja throwing blade). A folding bone may be used if an intricate model is made or it is hard to make a crease in paper. A model is made by folding paper carefully or tucking one end of the folded paper into a pocket. Because paper is folded continuously, a completed model can be much smaller than the unfolded paper. Typical origami models include cranes (folded paper crane or connected cranes), balloons, paper airplanes, shuriken, kabuto (a samurai war helmet) and yakkosan (a samurai's attendant).

There are Two Categories of Origami
The Japanese origami can be split into two categories. Origami like the paper crane model commonly seen is called yugi origami (play paper folding), and other origami like noshi (folded red and white paper) is called girei origami (ceremonial paper folding).

Girei Origami (Ceremonial paper folding)
As far as is shown in literature, the earliest reference to the girei origami was ocho and mecho (male and female butterfly models) in the haiku, 'butterflies in a beggar's dream are origami' composed by Saikaku IHARA in the 'Ittyuya Dokugin onsenku (4000 Haikus Recited Alone All Day and Night)" in 1680. The ocho and mecho, which are attached to the mouth of a sake bottle, are stylized bottle wrappings. Furthermore, the current noshi is a stylized noshi awabi (a thin strip of dried abalone wrapped in folded red and white paper), and an example of the girei origami. The girei origami is a part of a warrior family's manners and it would appear that it was developed mainly by the Ogasawara clan, the Ise clan and the Imagawa clan in the days of Yoshimitsu ASHIKAGA because "Tsutsumi no ki (record on wrapping)" written by Sadatake ISE in 1764 said that 'the folded paper models on the right were used during the time of the Kyoto shogun.'

Yugi Origami (Play paper folding)
The "Life of an Amorous Man" written by Saikaku IHARA in 1682 said that: One day he folded origami and gave it to a female servant, saying "This is a model of male and female birds in one." After making an origami flower, he attached it to the treetop and said, "This is the branch of a conjoined tree and I will give it to you." He never forgot anything concerning amorous affairs. The above-stated is the oldest record on the yugi origami. The "Hiden Senbazuru Origata" (Secret Folding of Thousand Cranes) published in 1797 was clearly targeted towards adults, which proves that adults as well as children enjoyed origami in those days. This is considered to be the oldest origami book in existence. The oldest origami in existence are origami models owned by the Moriwaki family. These models contain both the girei origami and the yugi origami, but the yugi origami was estimated to have been folded in the early 19th century.

Origami became International Culture
International origami groups were formed mainly by Akira YOSHIZAWA and Toshie TAKAHAMA from Japan, Robert Harbin from the UK and Liliane Oppenheimer and Samuel Randlett from the US, after which, origami spread throughout the world. Recently, intricate models have been made, starting with "Viva Origami" (created by Jun MEKAWA; authored by Kunihiko KASAHARA: ISBN 4387891165) published in 1983 and "Folding the Universe" (authored by Peter Engel) published in 1989. The technique of 'technical origami' developed by Jun MAEKAWA had a big influence on origami, and allowed for the rational designing of intricate models for the first time. When the Nippon Origami Association and Japan Origami Academic Society were established in Japan, origami groups formed in countries including the US and the UK, allowing origami fans to deepen exchanges with each other. Due to the development of the Internet, the speed of communication became faster and technical development has proceeded at an unprecedented speed.

Mathematics and Practical Applications of Origami
There are several mathematical challenges in the applications or study of origami. For example, flat-foldability, an issue as to whether or not a crease pattern can be folded into a two-dimensional model, is an example of a mathematical challenge. If the paper is flat, the Gaussian curvature is zero at any point on the surface. Therefore, the crease line is fundamentally a straight line with a Gaussian curvature of zero. The condition of this curvature cannot be, however, applied anymore to warped paper such as wet paper or paper wrinkled by a finger nail. The problem of rigid origami (whether or not a model can be folded like paper by using sheet metal with hinges in place of crease lines) is an important issue pertaining to practical use. For example, the Miura map fold allows a rigid body to be folded and has been used to fold large solar panels for space satellites. In addition, origami has been applied to fold air bags and stent grafts for medical use.

Origami was used in Japanese Ceremonial Event, such as Weddings
It is said that the origin of origami is the paper crafts of Japanese religion Shinto. Even now, you can find the trace of ancient shinto traditions that use folding pure white papers in the rituals. Pure white paper is believed to have the special power to charm against evil.
In Edo period (1603 - 1868), origami became widespread in the society along with the fall of the price of Japanese high-quality paper “Washi”. Then, “Noshi” culture arose, which were ceremonial folding papers attached to gifts, much like greeting cards in today.
In 1797, the book called “Hiden Senbazuru Orikata” (the secrets of how to fold the paper cranes) was published and it introduced 49 designs of paper crane for recreation. After that, many origami related books were published according to the rise of attention to origami culture.

Origami Absorbed European Napkin Folding Culture
In 1860s, Japan ends long period of isolation for modenization. In those days, Japan imported many cultures and knowledge from western countries, including kindergarten system. In the western kindergarten system, it is considered to be dangerous to use scissors to cut papers, so they ban the scissors even for origami. Also, western napkin folding culture was imported, then slowly integrated into Japanese origami culture.

Evolve to Modern Art with Mathematical Complexity
Akira Yoshikazu and Kosho Uchiyama were the two leading origami practitioners who invented modern origami works. Their inventions such as “wet-folding method” and “diagramming method” led the creation of totally new origami designs. These innovative origami methods created mathematically complex designs, therefore many origami practitioners started to study the mathematical properties of origami systematically. In recent years, this study produced many new and original origami art works.