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# Folding Processes
## **Why**
- Geometric folding is such a powerful tool when a physical object needs to be reconfigured.
- 2D processes that encodes a 3D shape are very interesting. https://www.economist.com/science-and-technology/2016/07/29/on-a-roll
- If you check in nature, there is folding everywhere.
## **Fast historic approach**
By the way, 6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra is an INCREDIBLE MIT class.
The way in which folding has been introduced in many aspects of our life has bee through the art of Origami, and it has been around us for a long time.
David Lister, as one of the most laureate members of the British Origami Society, holds in his
article Old European Origami that: “The subject of the History of Paper folding is beset by
enormous amount of conjecture which is based on little substantial fact. (...), And the reason of his words is because Origami uses a highly degradable material as its mean, making it impossible to exist any archeology of origami. Also, is an artistic expression that needs only hands and paper, so it leaves no traces. The only way to find facts, as David says, is finding evidences on books… And as a as a curiosity, the first evidence of paper folding in a book, is an illustration o from a British astronomer in a book called **De sphaera mundi**, written by Johennes Scabrosco in 1220 and published in 1490.
<img src="images/fold/fold1.png" alt="scheme" width = "800px">
And this image dated on the 19th century, is said to be the first book ilustation showing folding inside oriental society. At the 18th century the first Origami book called (Thousand Crane Folding) written by Akisato Rito.
<img src="images/fold/fold2.png" alt="scheme" width = "800px">
The most agreed version is that the history of Origami is strongly related to the History of paper. Due to this fact, it is widely
believed that paper folding is first made in China and would then spread over Japan and Middle
Asia. It is in Japan where folded paper figures appear in daily-life. At this point Origami models
are introduced in religious ceremonies named Shinto. Other easier patterns were developed and
transmitted orally.
### Stablishing the modern basics. *Akira Yoshizawa*
[Find here an awesome class revisiting the history of modern Origmi design](https://courses.csail.mit.edu/6.849/fall20/lectures/L04.html)
Father of modern Origami. For a long time, he was the first to start representing objects with *new origami models*, not only repeating the classic fold as a pray, but going deeper and trying to understand which are the rules of folding.
<img src="images/fold/fold3.png" alt="scheme" width = "800px">
He introduced the concept of wet folding, soaking the water in paper, shaping it and letting it dry.
But most importantly, he introduced a *language*. He created the standards we all follow now
<img src="images/fold/fold4.png" alt="scheme" width = "600px">
In the 20 century, after that huge legacy of new Yoshizawa designs, **math** irrupted drastically Origami. In less than a century, Origami passed from doing this:
<img src="images/fold/fold5.png" alt="scheme" width = "600px">
To this:
<img src="images/fold/fold6.png" alt="scheme" width = "600px">
Pockets of mathematicians interested in Origami were created but wasnt until 89 when there was the first Origami Science Math and Education conference when all them got together and this form of art *exploded*. All them, generated theorems and proof them.
## **4 basic laws rules for the vast majority of modern Origami.**
The form that a folded piece have in a unfolded state is its crease pattern. Those can not be drawn arbitrarely. They must follow 4 simple laws (theorems) that will ensure foldability. This is **not** universal as we will see, but cover whats called **flat foldable patterns**, which are the vast majority.
### 1. A triangulation of the polygons must construct a Hamiltonian Refinement [Arkin et all, 1996]
Some people simplify this as the 2 colorability pattern filling. You should be able to color all the poligons with alternative colors without having any edge sharing two same colored polygons.
<img src="images/fold/fold7.png" alt="scheme" width = "400px">
### 2. #mountains - # valleys around an edge = +-2. [Maekawa theorem 1986]
<img src="images/fold/fold8.png" alt="scheme" width = "400px">
### 3. θ<sub>1</sub> + θ<sub>3</sub> + ... + θ<sub>n-1</sub> = θ<sub>2</sub> + θ<sub>4</sub> + ... + θ<sub>n</sub> = 180 [Kawasaki 1989]
<img src="images/fold/fold9.png" alt="scheme" width = "400px">
### 4. Once folded, a sheet can never intersect an existing fold. [Hull 2001, Demaine & O'Rourke 2007]
<img src="images/fold/fold10.png" alt="scheme" width = "400px">
Results of that, state of the art Matematicians, Scientists, Engineer or Architects that are continuosly working is this field are:
- [Kōryō Miura](https://artsandculture.google.com/entity/g122_9hwg)
- [Tomohiro Tachi](https://origami.c.u-tokyo.ac.jp/~tachi/)
- [Erik Demaine](http://erikdemaine.org/)
- [Robert J.Lang](https://langorigami.com/)
- [Ekaterina Lukasheva](https://kusudama.me/)
# **Origami fields**
- [Kirigami](https://asmedigitalcollection.asme.org/mechanicaldesign/article-abstract/136/5/051011/375767/Manufacture-of-Arbitrary-Cross-Section-Composite?redirectedFrom=fulltext)
- [Thick origami](https://asmedigitalcollection.asme.org/mechanismsrobotics/article-abstract/12/2/021106/1072472/Rigidly-Foldable-Thick-Origami-Using-Designed?redirectedFrom=fulltext)
- [Rigid origami](https://iam.tugraz.at/workshop_rijeka/wp-content/uploads/2012/09/ThickRigidOrigami_tachi_5OSME.pdf)
- [Curve creased origami](https://erikdemaine.org/papers/Huffman_Origami5/paper.pdf)
# **Applications**
- Metamaterials
- [Tune Stiffness with Tachi - Miura cells](http://cba.mit.edu/docs/theses/19.09.calisch.pdf)
- [Tailored Electromagnetic Properties](https://www.nature.com/articles/s41427-018-0082-x)
- [Less material, more area, better mechanical properties](https://advances.sciencemag.org/content/1/8/e1500224)
- [Customized thermal expansion](https://bertoldi.seas.harvard.edu/files/bertoldi/files/boatti_adv_mat.pdf?m=1493950537)
- Tini tiny devices
- [Cornell micron bird, 60 micron wide, 30 atoms thick](https://news.cornell.edu/stories/2021/03/self-folding-nanotech-creates-worlds-smallest-origami-bird)
- [MEMS goes 3D. MIT nano Origami](https://news.mit.edu/2009/nano-origami-0224)
- Space
- [NASA starshade](https://exoplanets.nasa.gov/resources/2166/origami-starshade/)
- Robotics
- [Design, fabrication and control of origami robots](https://www.nature.com/articles/s41578-018-0009-8)
- [Miniature Origami Robot Self-folds, Walks, Swims, and Degrades](https://www.youtube.com/watch?v=f0CluQiwLRg&ab_channel=IEEESpectrum)
# **Design tools**
- [Tree maker](https://www.langorigami.com/article/treemaker)
- [Tomohiro Tachi legacy. Origamizer, Freeform Origami, Rigid Origami Simulator](https://tsg.ne.jp/TT/software/index.html)
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# **Simulation tools**
- [Incredible Origami simulator](https://origamisimulator.org/)
- [Merlin](https://paulino.ce.gatech.edu/conferences/papers/16Liu_merlin.pdf)
- [Beautifull abstraction as bars with rotations](https://www.sciencedirect.com/science/article/pii/S0020768317302408)
# **Manufacturing methods**
The problem is that some origami models takes like 10 hours to fold, and seconds to unfold.
3D shapes are encoded in 2D patterns. But this process needs to be parallel due to Rigid Origami is a parallel mechanism. Unfolding is very easy, folding is very hard.
That is why there is a bunch of proposals to self folding and not that much as Industrial Processes for folding.
- [Mechanical. Roll to roll processes](https://www.energy.gov/sites/prod/files/2015/02/f19/QTR%20Ch8%20-%20Roll%20To%20Roll%20Processing%20TA%20Feb-13-2015.pdf)
- [Mechanical. Rollers to arbitrary shaped HC](http://cba.mit.edu/docs/theses/19.09.calisch.pdf)
- [Self Folding. Programmable matters](https://www.pnas.org/content/107/28/12441)
- [Self folding. Polymer gel](https://www.princeton.edu/~akosmrlj/MAE545_S2017/lecture9_slides.pdf)
- [Self folding. Graphene folds](https://pubmed.ncbi.nlm.nih.gov/29295917/)
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