Mathematics of paper folding
The field of computational origami has also grown significantly since its inception in the 1990s with Robert Lang's TreeMaker algorithm to assist in the precise folding of bases.
[5] In 1936 Margharita P. Beloch showed that use of the 'Beloch fold', later used in the sixth of the Huzita–Hatori axioms, allowed the general cubic equation to be solved using origami.
No unduly complicated series of movements are required, and folded Miura-ori can be packed into a very compact shape.
[10] In 1985 Miura reported a method of packaging and deployment of large membranes in outer space,[11] and as early as 2012 this technique had been applied to solar panels on spacecraft.
[14] The first complete statement of the seven axioms of origami by French folder and mathematician Jacques Justin was written in 1986, but were overlooked until the first six were rediscovered by Humiaki Huzita in 1989.
[21][22] In 2002, Sarah-Marie Belcastro and Tom Hull brought to the theoretical origami the language of affine transformations, with an extension from
[26] In 2003, Jeremy Gibbons, a researcher from the University of Oxford, described a style of functional programming in terms of origami.
He characterizes fold and unfolds as natural patterns of computation over recursive datatypes that can be framed in the context of origami.
[28] In 2009, Alperin and Lang extended the theoretical origami to rational equations of arbitrary degree, with the concept of manifold creases.
Assigning a crease pattern mountain and valley folds in order to produce a flat model has been proven by Marshall Bern and Barry Hayes to be NP-complete.
[34] Some classical construction problems of geometry — namely trisecting an arbitrary angle or doubling the cube — are proven to be unsolvable using compass and straightedge, but can be solved using only a few paper folds.
[36] As a result of origami study through the application of geometric principles, methods such as Haga's theorem have allowed paperfolders to accurately fold the side of a square into thirds, fifths, sevenths, and ninths.
The accompanying diagram shows Haga's first theorem: The function changing the length AP to QC is self inverse.
For example: Haga's theorems are generalized as follows: Therefore, BQ:CQ=k:1 implies AP:BP=k:2 for a positive real number k.[37] The classical problem of doubling the cube can be solved using origami.
This construction is due to Peter Messer:[38] A square of paper is first creased into three equal strips as shown in the diagram.
[38][9] The angle CAB is trisected by making two folds: PP', parallel to the base, and QQ', halfway in between.
[41] Wet-folding origami is a technique evolved by Yoshizawa that allows curved folds to create an even greater range of shapes of higher order complexity.
[21][22] The fold-and-cut problem asks what shapes can be obtained by folding a piece of paper flat, and making a single straight complete cut.
The contest helped initialize a collective interest in developing universal models and tools to aid in origami design and foldability.
For example, a large enough piece of paper can be folded into any tree-shaped origami base, polygonal silhouette, and polyhedral surface.
[46] When universality results are not attainable, efficient decision algorithms can be used to test whether an object is foldable in polynomial time.
Computational intractability results show that there are no such polynomial-time algorithms that currently exist to solve certain folding problems.
The algorithm will be included in Origamizer, a free software for generating origami crease patterns that was first released by Tachi in 2008.
Users specify the desired shape or functionality and the software tool constructs the fold pattern and/or 2D or 3D model of the result.
[51] Computational origami has contributed to applications in robotics, engineering, biotechnology & medicine, industrial design.
[53] Robert Lang participated in a project with researchers at EASi Engineering in Germany to develop automotive airbag folding designs.
[54] In the mid-2000s, Lang worked with researchers at the Lawrence Livermore National Laboratory to develop a solution for the James Webb Space Telescope, particularly its large mirrors, to fit into a rocket using principles and algorithms from computational origami.
