Orgami

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 safe to say that most of its development occurred in Japan; however, there have also been less developed (but 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.

In the 1960s the art of origami began to spread out, first with modular origami and then with various movements developing, including kirigami. Origami is now an international art.

Paper and other materials

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.

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:

1. 

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 add up to 180 degrees, as do the even.
4. a sheet can never penetrate a fold

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 is a rigid fold that has been used to deploy large solar panel arrays for space satellites.