Origami

Origami (from oru meaning "folding", and kami meaning "paper") is the traditional Japanese art of paper folding. The goal of this art is to create a representation of an object using geometric folds and crease patterns preferably without the use of gluing or cutting the paper, and using only one piece of paper.

Origami only uses a small number of different folds, but they can be combined in a variety of ways to make intricate designs. The most well known form is probably the Japanese paper crane. In general, these designs begin with a square sheet of paper whose sides may be different colors or prints. Contrary to most popular belief, traditional Japanese origami, which has been practiced since the Edo era (1603–1867), has often been less strict about these conventions, sometimes cutting the paper during the creation of the design.

History
There is much speculation as to the origin of origami. It is generally believed that most of its modern developments occurred in Japan; however, there have also been independent paperfolding traditions in China, Germany, and Spain, among other places.

Origami had already become a significant aspect of Japanese ceremony by the Heian period of Japanese history. Samurai warriors would exchange gifts adorned with noshi, a sort of good luck token made of folded strips of paper. Origami butterflies were used during the celebration of Shinto weddings to represent the bride and groom.

Paper and other materials
Although almost any laminar material can be used for folding, the choice Normal copy paper with weights of 70–90 g/m² (19-24lb) can be used for simple folds, such as the crane and waterbomb. Heavier weight papers of 100 g/m² (approx. 25lb) or more can be wet-folded. This technique allows for a more rounded sculpting of the model, which becomes rigid and sturdy when it is dry.

Special origami paper, often also referred to as "kami" (Japanese for paper, among other things), is sold in prepackaged squares of various sizes ranging from 2.5 cm to 25 cm or more. It is commonly colored on one side and white on the other; however, dual coloured and patterned versions exist and can be used effectively for color-changed models. Origami paper weighs slightly less than copy paper, making it suitable for a wider range of models.

Foil-backed paper abbiee miller, just as its name implies, is a sheet of thin foil glued to a sheet of thin paper. Related to this is tissue foil, which is made by gluing a thin piece of tissue paper to kitchen aluminum foil. A second piece of tissue can be glued onto the reverse side to produce a tissue/foil/tissue sandwich. Foil-backed paper is available commercially, but not tissue foil; it must be handmade. Both types of foil materials are suitable for complex models.

Washi is the predominant origami paper used in Japan. Washi is generally tougher than ordinary paper made from wood pulp, and is used in many traditional arts. Washi is commonly made using fibers from the bark of the gampi tree, the mitsumata shrub (Edgeworthia papyrifera), or the paper mulberry but also can be made using bamboo, hemp, rice, and wheat.

Artisan papers such as unryu, lokta, hanji, gampi, kozo, saa, and abaca have long fibres and are often extremely strong. As these papers are floppy to start with, they are often backcoated or resized with methylcellulose or wheat paste before folding. Also, these papers are extremely thin and compressible, allowing for thin, narrowed limbs as in the case of insect models.

Paper money from various countries are also popular to create origami with, while some may call it "Moneygami," it is more accurately known as "Orikane." It is common to create the figure depicted on the note itself.

Thin tissue paper is used in hotel toilet-paper folding to indicate to guests that the bathroom has been recently cleaned.

Only paper is necessary to fold origami; however, some enthusiasts prefer to use a folding bone to sharpen creases while folding. Other folders grow certain nails long to aid with creasing instead of a folding bone.

Action Origami
Origami doesn't just cover still-lifes, it also covers moving objects; Origami can move in clever ways. Action origami includes origami that flies, requires inflation to complete, or, when complete, uses the kinetic energy of a person's hands, applied at a certain region on the model, to move another flap or limb. Some argue that strictly speaking, only the latter is really "recognized" as action origami. Action origami, first appearing with the traditional Japanese flapping bird, is quite common. One example is Robert Lang's instrumentalists; when the figures' heads are pulled away from their bodies, their hands will move, resembling the playing of music.

Mathematics of origami
The practice and study of origami encapsulates several subjects of mathematical interest. For instance, the problem of flat-foldability (whether a crease pattern can be folded into a 2-dimensional model) has been a topic of considerable mathematical study.

There are four mathematical rules for origami crease patterns: Significantly, paper exhibits zero Gaussian curvature at all points on its surface, and only folds naturally along lines of zero curvature. But the curvature along the surface of a non-folded crease in the paper, as is easily done with wet paper or a fingernail, is no longer subject to this constraint.
 * 1) crease patterns are two colorable
 * 2) at any vertex the number of valley and mountain folds always differ by two in either direction
 * 3) at any vertex, the sum of all the odd angles adds up to 180 degrees, as do the even.
 * 4) a sheet can never penetrate a fold

The problem of rigid origami ("if we replaced the paper with sheet metal and had hinges in place of the crease lines, could we still fold the model?") 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.

Technical origami
Technical origami, also known as origami sekkei, is a field of origami that has developed almost hand-in-hand with the field of mathematical origami. In the early days of origami, development of new designs was largely a mix of trial-and-error, luck and serendipity. With advances in origami mathematics however, the basic structure of a new origami model can be theoretically plotted out on paper before any actual folding even occurs. This method of origami design was developed by Robert Lang, Meguro Toshiyuki and others, and allows for the creation of extremely complex multi-limbed models such as many-legged centipedes, human figures with a full complement of fingers and toes, and the like.

The main starting point for such technical designs is the crease pattern (often abbreviated as 'CP'), which is essentially the layout of the creases required to form the final model. Although not intended as a substitute for diagrams, folding from crease patterns is starting to gain in popularity, partly because of the challenge of being able to 'crack' the pattern, and also partly because the crease pattern is often the only resource available to fold a given model, should the designer choose not to produce diagrams. Still, there are many cases in which designers wish to sequence the steps of their models but lack the means to design clear diagrams. Such origamists occasionally resort to the Sequenced Crease Pattern (abbreviated as SCP) which is a set of crease patterns showing the creases up to each respective fold. The SCP eliminates the need for diagramming programs or artistic ability while maintaining the step-by-step process for other folders to see. Another name for the Sequenced Crease Pattern is the Progressive Crease Pattern (PCP).

Paradoxically enough, when origami designers come up with a crease pattern for a new design, the majority of the smaller creases are relatively unimportant and added only towards the completion of the crease pattern. What is more important is the allocation of regions of the paper and how these are mapped to the structure of the object being designed. For a specific class of origami bases known as 'uniaxial bases', the pattern of allocations is referred to as the 'circle-packing'. Using optimization algorithms, a circle-packing figure can be computed for any uniaxial base of arbitrary complexity. Once this figure is computed, the creases which are then used to obtain the base structure can be added. This is not a unique mathematical process, hence it is possible for two designs to have the same circle-packing, and yet different crease pattern structures.

As a circle encloses the minimum amount of area for a given perimeter, circle packing allows for maximum efficiency in terms of paper usage. However, other polygonal shapes can be used to solve the packing problem as well. The use of polygonal shapes other than circles is often motivated by the desire to find easily locatable creases (such as multiples of 22.5 degrees) and hence an easier folding sequence as well. One popular offshoot of the circle packing method is box-pleating, where squares are used instead of circles. As a result, the crease pattern that arises from this method contains only 45 and 90 degree angles, which makes for easier folding.

Origami Tessellations
This branch of origami is one that has grown in popularity recently, but has an extensive history. Tessellations refer to the tiling of the plane where a collection of 2 dimensional figures fill a plane with no gaps or overlaps. Origami tessellations are tessellations made from a flat material, most often paper, but it can be from anything that holds a crease. The history of costuming includes tessellations done in fabric that are recorded as far back as the Egyptian Tombs.

Fujimoto was an early Japanese origami master who published books that included origami tessellations and in the 1960's there was a great exploration of tessellations by Ron Resch. Chris Palmer is an artist who has extensively explored tessellations and has found ways to create detailed origami tessellations out of silk. Robert Lang and Alex Bateman are two designers who use computer programs to design origami tessellations. The first American book on origami tessellations was just published by Eric Gjerde and the field has been expanding rapidly. There are numerous origami tessellation artists including Eric Gjerde (U.S.), Polly Verity (Scotland), Joel Cooper (U.S.), Christine Edison (U.S), Ray Schamp (U.S.), Roberto Gretter (Italy), Goran Konjevod (U.S),and Christiane Bettens (Switzerland) that are showing works that are both geometric and representational.

Space flight
There may soon be an origami airplane launched from space. A prototype passed a durability test in a wind tunnel on March 2008, and Japan's space agency adopted it for feasibility studies.

Authors of books about origami

 * Akira Yoshizawa - reinvented modern origami. Created the modern repertoire of folding symbols
 * Robert J. Lang - Author of many Origami books including the new benchmark Origami Design Secrets
 * Peter Engel - influential origami artist and theorist
 * Tomoko Fuse (布施 知子) - famous for boxes and unit origami
 * Robert Harbin - popularised origami in Britain; also presented a series of short programmes entitled Origami, made by Thames Television for ITV
 * Eric Joisel - French wet-folder renowned for his life-like masks
 * Kunihiko Kasahara - devised a standardized method for creating many modular polyhedra
 * Duy Nguyen - master artist who wrote many origami books
 * John Montroll - probably the most prolific Western artist and author of over 16 books on origami
 * Nick Robinson - origami artist and author of over a dozen books on origami
 * Toshikazu Kawasaki - Japanese mathematician famous for his Iso-area folding theory and his many geometric folds, including Kawasaki's "Rose"
 * Makoto Yamaguchi
 * Jeremy Shafer - California origamian entertainer who folds whimsical designs
 * Satoshi Kamiya - One of the youngest geniuses of the origami field
 * Nicolas Terry - French artist known for his unique, cartoonish style
 * Issei Yoshino - famed for his multimodular Tyrannosaurus and Triceratops skeletons 
 * Marc Kirschenbaum - known for his instrumentalist designs
 * Didier Boursin
 * Florence Temko - a pioneer in spreading origami in the United States, is the most prolific author on this subject. With 55 books to her credit on paperarts and folk crafts, she has been a strong influence on interesting beginners in the art of paperfolding.
 * David Shall - American designer, whose diagrams are completely hand drawn. Perhaps best known for his model of the Statue of Liberty
 * Eric Gjerde - American designer of origami tessellations. His book is Origami Tessellations: Awe-Inspiring Geometric Designs
 * Linda Wright - California author of Toilet Paper Origami, an elegant, charming and quirky origami for hospitality.
 * Joel Stern - Author of Jewish Holiday Origami, a collection of models celebrating Jewish culture, and Animated Origami Faces, movable designs made from U.S. letter-size paper.