User:ArnavPrasadChicago/Mathematics of paper folding

Mathematics of Paper Folding
The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper folds to solve up-to cubic mathematical equations.

Computational origami is a recent branch of computer science that is concerned with studying algorithms that solve paper-folding problems. 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. Computational origami results either address origami design or origami foldability. In origami design problems, the goal is to design an object that can be folded out of paper given a specific target configuration. In origami foldability problems, the goal is to fold something using creases of an initial configuration. Results in origami design problems have been more accessible than in origami foldability problems.

Computational Origami
Computational origami is a branch of computer science that is concerned with studying algorithms for solving paper-folding problems. In the early 1990s, origamists participated in a series of origami contests called the Bug Wars in which artists attempted to out-compete their peers by adding complexity to their origami bugs. Most competitors in the contest belonged to the Origami Detectives, a group of acclaimed Japanese artists. Robert Lang, a research-scientist from Stanford University and the California Institute of Technology, also participated in the contest. The contest helped initialize a collective interest in developing universal models and tools to aid in origami design and foldability.

Research
Paper-folding problems are classified as either origami design or origami foldability problems. There are predominantly three current categories of computational origami research: universality results, efficient decision algorithms, and computational intractability results. A universality result defines the bounds of possibility given a particular model of folding. For example, a large enough piece of paper can be folded into any tree-shaped origami base, polygonal silhouette, and polyhedral surface. When universality results are not attainable, efficient decision algorithms can be used to test whether an object is foldable in polynomial time. Certain paper-folding problems do not have efficient algorithms. Computational intractability results show that there are no such polynomial-time algorithms that currently exist to solve certain folding problems. For example, it is NP-hard to evaluate whether a given crease pattern folds into any flat origami.

In 2017, Erik Demaine of the Massachusetts Institute of Technology and Tomohiro Tachi of the University of Tokyo published a new universal algorithm that generates practical paper-folding patterns to produce any 3-D structure. The new algorithm built upon work that they presented in their paper in 1999 that first introduced a universal algorithm for folding origami shapes that guarantees a minimum number of seams. The algorithm will be included in Origamizer, a free software for generating origami crease patterns that was first released by Tachi in 2008.

Software & Tools
There are several software design tools that are used for origami design. Users specify the desired shape or functionality and the software tool constructs the fold pattern and/or 2D or 3D model of the result. Researchers at the Massachusetts Institute of Technology, Georgia Tech, University of California Irvine, University of Tsukuba, and University of Tokyo have developed and posted publicly available tools in computational origami. TreeMaker, ReferenceFinder, OrigamiDraw, and Origamizer are among the tools that have been used in origami design.

There are other software solutions associated with building computational origami models using non-paper materials such as Cadnano in DNA origami.

Applications
Computational origami has contributed to applications in robotics, biotechnology & medicine, industrial design.

Robert Lang, one of the pioneers of computational origami, participated in a project with researchers at EASi Engineering in Germany to develop automotive airbag folding designs. 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.

In 2014, researchers at the Massachusetts Institute of Technology, Harvard University, and the Wyss Institute for Biologically Inspired Engineering published a method for building self-folding machines and credited advances in computational origami for the project's success. Their origami-inspired robot was reported to fold itself in 4 minutes and walk away without human intervention, which demonstrated the potential for autonomous self-controlled assembly in robotics.

Other applications include DNA origami and RNA origami, folding of manufacturing instruments, and surgery by tiny origami robots.

Applications of computational origami have been featured by various production companies and commercials. Lang famously worked with Toyota Avalon to feature an animated origami sequence, Mitsubishi Endeavor to create a world entirely out of origami figures, and McDonald's to form numerous origami figures from cheeseburger wrappers.