Origami
) is the art of paper folding, which is often associated with Japanese culture. In modern usage, the word "origami" is used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a finished sculpture through folding and sculpting techniques. Modern origami practitioners generally discourage the use of cuts, glue, or markings on the paper. Origami folders often use the Japanese word "" to refer to designs which use cuts. 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. History. Distinct paperfolding traditions arose in Europe, China, and Japan which have been well-documented by historians. These seem to  have been mostly separate traditions, until the 20th century. In China, traditional funerals often include the burning of folded paper, most often representations of gold nuggets (yuanbao). The practice of burning paper representations instead of full-scale wood or clay replicas dates from the Song Dynasty (905–1125 CE), though it is not clear how much folding was involved. In Japan, the earliest unambiguous reference to a paper model is in a short poem by Ihara Saikaku in 1680 which mentions a traditional butterfly design used during Shinto weddings. Folding filled some ceremonial functions in Edo period Japanese culture; noshi were  attached to gifts, much like greeting cards are used today. This developed into a form of entertainment; the first two instructional books published in Japan are clearly recreational. In Europe, there was a well-developed genre of napkin folding, which flourished during the 17th and 18th centuries. After this period, this genre declined and was mostly forgotten; historian Joan Sallas attributes this to the introduction of porcelain, which replaced complex napkin folds as a dinner-table status symbol among nobility. However, some of the techniques and bases associated with this tradition continued to be a part of European culture; folding  was a significant part of Friedrich Fröbel's "Kindergarten" method, and the designs published in connection with his curriculum are stylistically similar to the napkin fold repertoire. Another example of early origami in Europe is the "pajarita," a stylized bird whose origins date from at least the nineteenth century. When Japan opened its borders in the 1860s, as part of a modernization strategy, they imported Fröbel's Kindergarten system—and with it, German ideas about paperfolding. This included the ban on cuts, and the starting shape of a bicolored square. These ideas, and some of the European folding repertoire, were  integrated into the Japanese tradition. Before this, traditional Japanese sources use a variety of starting shapes, often had cuts; and if they had color or markings, these were added after the model was folded. In the early 1900s, Akira Yoshizawa, Kosho Uchiyama, and others began creating and recording original origami works. Akira Yoshizawa in particular was responsible for a number of innovations, such as wet-folding and the Yoshizawa–Randlett diagramming system, and his work inspired a renaissance of the art form. During the 1980s a number of folders started systematically studying the mathematical properties of folded forms, which led  to a rapid increase in the complexity of origami models. Starting in the late 20th century, there has been a renewed interest in understanding the behavior of folding matter, both artistically and scientifically. The "new origami," which distinguishes it from old craft practices, has had a rapid evolution due to the contribution of computational mathematics and the development of techniques such as "box-pleating", "tessellations" and "wet-folding". Artists like Robert J. Lang, Erik Demaine, Sipho Mabona, Giang Dinh, Paul Jackson, and others, are frequently cited for advancing new applications of the art. The computational facet and the interchanges through  social networks, where new techniques and designs are introduced, have raised the profile of origami in the 21st century. Techniques and materials. Techniques. Many origami books begin with a description of basic origami techniques which are used to construct the models. This includes simple diagrams of basic folds like valley and mountain folds, pleats, reverse folds, squash folds, and sinks. There are also standard named bases which are used in a wide variety of models, for instance the bird base is an intermediate stage in the construction of the flapping bird. Additional bases are the preliminary base (square base), fish  base, waterbomb base, and the frog base. Origami paper. Almost any laminar (flat) material can be used for folding; the only requirement is that it should hold a crease. Origami paper, often referred to as "kami" (Japanese for paper), is sold in prepackaged squares of various sizes ranging from 2.5 cm (1 in) to 25 cm (10 in) 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. Normal copy paper with weights of 70–90 g/m (19–24 lb) can be used for simple folds, such as the crane and waterbomb. Heavier weight papers of 100 g/m (approx. 25 lb) 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. Foil-backed paper, 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 aluminium 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. Artisan papers such as unryu, lokta, hanji, gampi, kozo, saa, and abaca have long fibers 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 is also popular to create origami with; this is known variously as Dollar Origami, Orikane, and Money Origami. Tools. It is common to fold using a flat surface, but some folders like doing it in the air with no tools, especially when displaying the folding. Some folders believe that no tool should be used when folding. However a couple of tools can help especially with the more complex models. For instance a bone folder allows sharp creases to be made in the paper easily, paper clips  can act as extra pairs of fingers, and tweezers can be used to make small folds. When making complex models from origami crease patterns, it can help to use a ruler and ballpoint embosser to score the creases. Completed models can be sprayed so that they keep their shape better, and a spray is needed when wet folding. Types. Action origami. In addition to the more common still-life origami, there are also moving object designs; origami can move. 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. Modular origami. Modular origami consists of putting a number of identical pieces together to form a complete model. Often the individual pieces are simple, but the final assembly may be more difficult . Many modular origami models are decorative folding balls such as kusudama, which differ from classical origami in that the pieces may be held together using thread or glue. Chinese paper folding, a cousin of origami, includes a similar style called golden venture folding where large numbers of pieces are put together to create elaborate models. This style is most commonly known as "3D origami". However, that name did not appear until Joie Staff published a series of books titled "3D Origami", "More 3D Origami", and "More and More 3D Origami". This style originated from some Chinese refugees while they were  detained in America and is also called Golden Venture folding from the ship they came on. Wet-folding. Wet-folding is an origami technique for producing models with gentle curves rather than geometric straight folds and flat surfaces. The paper is dampened so it can be moulded easily, the final model keeps its shape when it dries. It can be used, for instance, to produce very natural looking animal models. Size, an adhesive that is crisp and hard when dry, but dissolves in water when wet and becoming soft and flexible, is often applied to the paper either at the  pulp stage while the paper is being formed, or on the surface of a ready sheet of paper. The latter method is called external sizing and most commonly uses Methylcellulose, or MC, paste, or various plant starches. Pureland origami. Pureland origami adds the restrictions that only simple mountain/valley folds may be used, and all folds must have straightforward locations. It was developed by John Smith in the 1970s to help inexperienced folders or those with limited motor skills. Some designers also like the challenge of creating within the very strict constraints. Origami tessellations. Origami tessellation is a branch that  has grown in popularity after 2000. A tessellation is a collection of figures filling a plane with no gaps or overlaps. In origami tessellations, pleats are used to connect molecules such as twist folds together in a repeating fashion. During the 1960s, Shuzo Fujimoto was the first to explore twist fold tessellations in any systematic way, coming up with dozens of patterns and establishing the genre in the origami mainstream. Around the same time period, Ron Resch patented some tessellation patterns as part of his explorations into kinetic sculpture and developable surfaces, although his work was not known by the  origami community until the 1980s. Chris Palmer is an artist who has extensively explored tessellations after seeing the Zilij patterns in the Alhambra, 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 create origami tessellations. The first international convention devoted to origami tessellations was hosted in Brasília (Brazil) in 2006, and the first instruction book on tessellation folding patterns was published by Eric Gjerde in 2008. Since then, the field has grown very quickly. Tessellation artists include Polly Verity (Scotland); Joel Cooper, Christine Edison , Ray Schamp and Goran Konjevod from the USA; Roberto Gretter (Italy); Christiane Bettens (Switzerland); Carlos Natan López (Mexico); and Jorge C. Lucero (Brazil). Kirigami. Kirigami is a Japanese term for paper cutting. Cutting was often used in traditional Japanese origami, but modern innovations in technique have made the use of cuts unnecessary. Most origami designers no longer consider models with cuts to be origami, instead using the term Kirigami to describe them. This change in attitude occurred during the 1960s and 70s, so early origami books often use cuts, but for the most part they have disappeared from the modern  origami repertoire; most modern books don't even mention cutting. Strip folding. Strip folding is a combination of paper folding and paper weaving. A common example of strip folding is called the Lucky Star, also called Chinese lucky star, dream star, wishing star, or simply origami star. Another common fold is the Moravian Star which is made by strip folding in 3-dimensional design to include 16 spikes. Teabag folding. Teabag folding is credited to Dutch artist Tiny van der Plas, who developed the technique in 1992 as a papercraft art for embellishing greeting cards. It uses small square pieces  of paper (e.g., a tea bag wrapper) bearing symmetrical designs that are folded in such a way that they interlock and produce a three-dimensional version of the underlying design. The basic kite fold is used to produce rosettes that are a 3 dimensional version of the 2D design. The basic rosette design requires eight matching squares to be folded into the 'kite' design. Mathematics teachers find the designs very useful as a practical way of demonstrating some basic properties of symmetry. Mathematics and technical origami. Mathematics and practical applications. 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. A number of technological advances have come from insights obtained through paper folding. For example, techniques have been developed for the deployment of car airbags and stent implants from a folded position. 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. Origami can be used to construct various geometrical designs not possible with compass and straightedge constructions. For instance paper folding may be used for angle trisection and doubling the cube. Technical origami. Technical origami, known in Japanese as , is an origami design approach in which the model is conceived as an engineered crease pattern, rather than developed through trial-and-error. With advances in origami mathematics, 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 crease pattern is a layout of the creases required to form the structure of the model. 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 model. 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. By opening up a folded model, you can observe the structures that comprise it; the study of these structures led to a number of crease-pattern-oriented design approaches The pattern of allocations is referred to as the 'circle-packing' or 'polygon-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 maximum 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 often makes for a more direct folding sequence. Origami-related computer programs. A number of computer aids to origami such as TreeMaker and Oripa, have been devised. TreeMaker allows new origami bases to be designed for special purposes and Oripa tries to calculate the folded shape from the crease pattern. Ethics and copyright. Copyright in origami designs and the use of models has become an increasingly  important issue in the origami community, as the internet has made the sale and distribution of pirated designs very easy. It is considered good etiquette to always credit the original artist and the folder when displaying origami models. It has been claimed that all commercial rights to designs and models are typically reserved by origami artists; however, the degree to which this can be enforced has been disputed. Under such a view, a person who folds a model using a legally obtained design could publicly display the model unless such rights were specifically reserved, whereas folding a design for money  or commercial use of a photo for instance would require consent. The Origami Authors and Creators group was set up to represent the copyright interests of origami artists and facilitate permissions requests. However, a court in Japan has asserted that the folding method of an origami model "comprises an idea and not a creative expression, and thus is not protected under the copyright law". Further, the court stated that "the method to folding origami is in the public domain; one cannot avoid using the same folding creases or the same arrows to show the direction in which to fold  the paper". Therefore, it is legal to redraw the folding instructions of a model of another author even if the redrawn instructions share similarities to the original ones, as long as those similarities are "functional in nature". The redrawn instructions may be published (and even sold) without necessity of any permission from the original author. Gallery. These pictures show examples of various types of origami.