Origami (from ori meaning "folding", and kami meaning "paper") is the ancient Japanese art of paper folding. The goal of this art is to create a given result using geometric folds and crease patterns preferably without the use of gluing or cutting the paper medium. "Origami" nowadays refers to all types of paper folding, even those of non-Asian origin.
Origami only uses a small number of different folds, but they can be combined in a variety of ways to make intricate designs. 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.
The origin of the art began as Chinese Paper Folding. The Japanese origin began in the 6th century when Buddhist monks from China carried paper to Japan. The first Japanese origami is dated from this period. 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.
In the 1960's the art of origami began to spread out, first with modular origami and then with various movements developing, including the kirikomi.
Origami doesn't just cover still-lifes it covers real life objects; Origami can move in clever ways. Action origami includes origami that flies, requires inflation to complete, or, when complete, utilizes the kinetic energy of your hands applied at a certain region on the model and transfers it through an internal mechanism to move another flap or limb. Strictly speaking only the latter is really "recognized" as action origami. Action origami, first appearing with the traditional Japanese flapping bird, is quite common with Robert Lang's instrumentalists; when the figures' heads are pulled away from their bodies, their hands will move, resembling to play music.
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.
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.
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, also known as origami sekkei (折り紙設計?), 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., origami sekkei
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.