* This page was written in March 2021. The information on this page is as at the time it was written.
In the late 1900s, various Origami artists created their original works.
They were so creative that they generated new techniques and new designs which had never been imagined.
Can Origami create any shapes that we like?
Or, does Origami have a shape that cannot be created? If so, what is it like?
Some Origami artists came up with such questions, and they gradually discovered the mathematical principles of Origami.
Good examples of those principles are Kawasaki's theorem and Maekawa's theorem.
Each of them was named after the discoverer, Kawasaki Toshikazu (川崎敏和) and Maekawa Jun (前川淳), both of whom are Origami artists and researchers. The theorems are as follows.
Kawasaki's theorem:
The patterns of crease lines meeting at one vertex are flat-foldable if and only if the alternating sum of the angles is 0.
Maekawa's theorem:
At every vertex on a flat-foldable Origami crease pattern, the numbers of mountain and valley fold always differ by 2.
| [7-0] a flat-foldable pattern |
Findings like these theorems attracted not only Origami artists but also mathematicians.
Nowadays, various researches on Origami mathematics are carried on, and many discoveries are applied to various scientific fields.
cf. Takehisa, M., & Jun, M. (2017). Mathematics of Flat Origami. Nihon Ōyō Sūri Gakkai ronbunshi, 27(4), 333-353 *in Japanese. An abstract in English is available.
cf. Hidefumi, K. (2017). An Application of a theorem of alternatives to origami. Journal of the Operations Research Society of Japan. 60(3), 393-399 *in English
| [Image] | |
| 7-1 | Kawasaki's theorem.jpg by David Eppstein on Wikimedia Commons / CC BY-SA 3.0 (Retrieved 30 June, 2021) |
Even if one sheet of paper is thin and weak, after properly folded, it gets the power to stand up by itself. Moreover, when it adds more crease lines, it gets much stronger.
Just by folding, you can provide new properties for a sheet of paper.
Also, just by folding, you can control the movement of a sheet of paper.
In that sense, Origami attracts many engineers and architects, and they are carrying on various researches.
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[7-1] Miura-ori |
Miura-ori can be one of the examples of those researches.
This was invented by Miura Koryo (三浦公亮), a professor emeritus at the University of Tokyo.
Miura-ori is the geometrical crease pattern, by which you can make a sheet of paper very easy to open and close.
[7-1] shows the motion of a sheet with Miura-ori. As you can see, by Miura-ori, you can open and close the sheet in just one motion, and you can fold the sheet into a very compact shape.
Because of these advantages, Miura-ori is applied to various items, from a road map for tourists to a solar panel for a spacecraft like the Space Flyer Unit launched in 1995 by JAXA.
Researches by Tach Tomohiro (舘知宏, The University of Tokyo) are also an example. He analyzes the structure of Origami with consideration of qualities of materials and applies the structure to architectural design in practice.
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[7-2] Stiff and Flexible Sandwich Panel |
UTokyo OCW provides movies of the lectures by him, which show you what mathematical principles are hidden behind Origami and how researchers are carrying on their researches on Origami.
[UTokyo Open Course Ware]
Held on 2016 Mathematical Science Developing from Figures (Global Focus on Knowledge) *in Japanese
- #4 Computational Origami and Shapes
- #5 Mechanisms of Origami
| [Image] | |
| 7-1 | Miura-ori.gif by MetaNest on Wikimedia Commons / CC BY-SA 3.0 (Retrieved 8 March, 2021) |
| 7-2 | Stiff and Flexible Sandwich Panel by Tomohiro Tachi on Flicr / CC BY-NC 2.0 (Retrieved 8 March, 2021) |
| [Reference] | |
| - | ミウラ折りとは on 株式会社miuraori-lab (Retrieved 8 March, 2021. In Japanese. The original site has expired. This is the link to the Internet Archive.) |
The technology to fold effectively can be applied to medical science.
Shigetomi Kaori (繁富香織, Hokkaido University) develops the cell origami folding technique.
This technique is a method to form the 3D microstructure of cells by using the power of cells' self-folding.
Co-cultured cells usually form a 2D structure like a sheet.
By the cell origami technique, a 3D structure can be constructed from a sheet of cells, which enables researchers to make a simulated structure much similar to real organisms.
This technique is expected to be applied to medical or biological science.
cf. Kaori, S. (2017). 細胞折り紙と医療. Seibutsu Kogaku Kaishi. 94(5), 282-283 *in Japanese
By the way, is human the only creature that has a technique to fold? The answer is No.
The research group of Saito Kazuya (斉藤一哉, Kyushu University) analyzed complex crease patterns of the wings of an insect, earwig. Using 3D micro CT imaging technology, they analyzed the patterns by Origami geometry, and they revealed that the patterns are produced by simple geometrical rules, which can be applied to architecture and engineering.
The research group used the same method for the analysis of a fossil of a 280-million-year-old earwig relative, and they showed that the same geometrical rules as today's earwigs can be applied to the ancient earwig relatives. This is an example that origami geometry reveals one aspect of evolution.
cf. Kyushu University. (14 July, 2020). Design of insect-inspired fans offers wide-ranging applications. [News Release] *in English
Origami also attracts computer scientists.
When you fold a sheet of paper, you make some crease lines on it. Those lines construct geometrical patterns, which are called crease patterns.
There must be a logical relationship between crease patterns and the work you make, and such a relationship must be useful to design new origami works logically.
In that sense, there are now various computer programs developed to help Origami artists design their works.
As mentioned on THIS page, Robert J. Lang developed 'TreeMaker'.
This is the program that supports you to get valid crease patterns.
When you input the basic structure of the work you want to make, TreeMaker generates the crease patterns as an output, and you can make a new Origami work (as long as you can fold a sheet of paper as the crease patterns).
cf. TREEMAKER（Robert J. Lang Origami）
In turn, there is a program that generates Origami works as outputs.
Mitani Jun (三谷純, The University of Tsukuba) developed the program 'ORIPA'.
This program has many useful functions for you to draw crease patterns on a sheet.
If you input crease patterns of the work that is flat-foldable, ORIPA can simulate what the shape is like when it is folded properly.
In addition to ORIPA, Mitani developed other programs, for example, 'ORI-REVO', the program which generates 3D works based on a surface of revolution, and 'ORI-REF', the program with which you can design works with curved lines.
cf. Origami Applications by Jun Mitani（Jun MITANI）
Origami attracts mathematicians. Their findings are applied to computer science. Finally, their programs support Origami artists' creativity...
Today's Origami is right in the process of evolution between art and science.
To share the scientific findings on Origami, scientists hold the convention, International Meeting of Origami Science, Mathematics, and Education (OSME), every few years, and each time they published the proceedings.
