Editing Origami (section)

==Mathematics and technical origami==

===Mathematics and practical applications===
[[Image:Origami spring.jpg|thumb|''[[Spring (device)|Spring]] Into Action'', designed by Jeff Beynon, made from a single rectangular piece of paper<ref>[http://www1.ttcn.ne.jp/~a-nishi/ The World of Geometric Toy], ''[http://www1.ttcn.ne.jp/~a-nishi/spring/z_spring.html Origami Spring]'', August, 2007.</ref>]]

{{Main|Mathematics of paper folding}}
The practice and study of origami encapsulates several subjects of [[mathematics|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 [[airbag]]s and [[stent]] implants from a folded position.<ref>Cheong Chew and Hiromasa Suziki, ''Geometrical Properties of Paper Spring'', reported in Mamoru Mitsuishi, Kanji Ueda, Fumihiko Kimura, ''Manufacturing Systems and Technologies for the New Frontier'' (2008), p. 159.</ref>

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 {{Nihongo|'''origami sekkei'''|折り紙設計}}, 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 J. Lang|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.<ref>{{cite web|url=https://langorigami.com/article/treemaker/|title=TreeMaker|access-date=29 October 2022}}</ref> 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.<ref>{{cite book |title=Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education |editor= Patsy Wang-Iverson |editor2=Robert James Lang |editor3=Mark Yim |isbn=978-1-56881-714-9 |year=2010 |publisher=CRC Press |pages=335–370}}</ref> TreeMaker allows new origami bases to be designed for special purposes<ref>{{cite web |url=https://langorigami.com/article/treemaker/ |title=TreeMaker |first=Robert |last=Lang |access-date=October 29, 2022}}</ref> and Oripa tries to calculate the folded shape from the crease pattern.<ref>{{cite web |url=https://mitani.cs.tsukuba.ac.jp/oripa/ |title=ORIPA: Origami Pattern Editor |first=Jun |last=Mitani |access-date=October 29, 2022 }}</ref>