SIAM News Blog

Reflecting on the Science, Technology, and Artistry of Origami Engineering

By Ichiro Hagiwara

Origami engineering encompasses engineering, artistic, and geometrical pursuits. The engineering aspect specifically comprises three pillars: origami design, design manufacturing for stiff origami structures, and design manufacturing for deployable and foldable structures (see Figure 1).

Washi is considered to be the world's first foldable paper. Because this paper was developed in Japan, it is perhaps inevitable that origami was subsequently born and nurtured there as well. Throughout the peaceful and prosperous Edo period (1603-1868 CE), Japanese culture and art forms such as haiku (poetry), kabuki (theatre), noh (dance-drama), and ogi (folding fans) became increasingly popular. Although origami was long considered to be a precious type of art rather than a means of economic profit, that outlook changed during World War II when British engineers developed a mass production method for honeycomb structures—inspired by Japanese Tanabata decorations—that grew into a trillion-yen industry. Honeycomb structures possess a high bending stiffness given their weight, find applications in rockets and bullet trains, and ultimately inspired the myriad possibilities of origami engineering. Taketoshi Nojima of Kyoto University originally proposed the field of origami engineering in 2002, and I promoted this research area by organizing the Origami Engineering Research Group within the Japan Society for Industrial and Applied Mathematics  and the Japan Society of Mechanical Engineers.

In 2006, Nojima incorporated traditional Japanese kirigami—a variation of origami that allows for cutting as well as folding—to conceptualize curved honeycombs [7], which were impossible to create with existing British manufacturing methods. In 2014, Kazuya Saito of the University of Tokyo proposed a new manufacturing method that changes the patterns of slits or polygonal lines to respectively produce wing-sharp or eave-sharp honeycombs [10]. These kirigami honeycombs are now one of the hottest trends in origami engineering because they offer a treasure trove of manufacturing for metamaterials. Much like origami, kirigami has since become an international phenomenon. And despite the challenges of fashioning arbitrary-shaped kirigami honeycombs with a single connection, we were able to successfully build such a structure with only one honeycomb in 2023 [1].

Figure 1. Three pillars of the industrial engineering component of origami engineering. Figure courtesy of Ichiro Hagiwara.

The notion of biomimetic origami arose from the knowledge that many plants grow new parts at specific angles, thus creating a spiral pattern. This pattern consists of right and left equal-angle spirals; the number of spirals is a combination of two consecutive numbers in the Fibonacci sequence. Based on this observation, Nojima developed splendid origami structures such as cylindrical shells, conical shells, and circular membrane structures [6]. 

Although it is possible to obtain a development view that provides a schematic of the mountain and valley folds for all of these origami structures, doing so is more difficult for conical shell and circular membrane origami structures than cylindrical origami structures. The relational expressions between angles of intersecting points in conical shells and circular membranes decide the conditions under which we can create a foldable three-dimensional (3D) structure without contradiction from a development view; this relationship inspired us to employ conformal mapping [5]. We can then generate various origami winding models by utilizing vortex conformal mappings of the cylindrical origami structure. In fact, world-famous fashion designer Issey Miyake even used this origami winding model to create a folding dress.

We have applied these deployable and foldable properties of origami structure in the context of a vehicle crash energy absorber. As shown by the red dotted line in Figure 2, the initial peak load of a conventional vehicle absorber is high and can potentially lead to passenger injury; even the amount of deformation remains at only about 70 percent of its length. To solve this problem, we propose a reversed spiral origami (RSO) structure as a new energy-absorbing material [3]. The black line in Figure 2 shows that the RSO’s load value is almost constant without much change from deformation, and the amount of deformation reaches roughly 90 percent of its length due to the folding condition. In addition, we are researching materials that can absorb high amounts of energy during impact and are manufacturable at a lower cost than the current architecture [12].

Figure 2. Graphical depiction of a vehicle crash energy absorber. Because the initial peak load of a conventional vehicle absorber is high (red dotted line) and can lead to passenger injury, we propose a new energy-absorbing material in the form of a reversed spiral origami (RSO) structure (black line). Figure courtesy of Ichiro Hagiwara.
We have also explored applications to foldable polyethylene terephthalate (PET) bottles [4]. In 2007, a major Japanese newspaper ran an article—entitled “Origami Engineering Goes Global: From PET Bottles to Space” [8]—that promised the forthcoming creation of a folded PET bottle (though we did not ultimately succeed in crafting such a bottle until 2019). The difficulty was that the bottle’s springback to full size and original height deployed even when the structure was folded. By adding grooves within which the folded part could fit, we finally obtained the desired folded PET bottle. We have used the deployable and foldable properties of origami in many other types of applications as well, including thick plate folding box origami hydraulic dampers and vibration isolation and suspension systems.

Now let’s consider origami design, which seeks to reproduce objects with origami. In 2007, Tomohiro Tachi of the University of Tokyo developed the Origamizer: a software that generates development diagrams that allow one to fold complex polyhedral shapes from a single sheet of paper without any gaps (Tachi has since worked with Erik Demaine of the Massachusetts Institute of Technology to further extend this software). Meanwhile, we built a system that uses reverse engineering to create origami mountain and valley fold lines from photographs and development drawings. This concept is also viable for 3D additive printers; using an origami robot, we could potentially use an origami printer to automatically obtain the origami design [2]. Baby diapers are one example of a potential lucrative application of this method.

As part of the American Society of Mechanical Engineers’ 2016 Student Mechanism and Robot Design Competition—which took place during the ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference—we created the first origami robot that could glue multiple components together. Developing an origami robot is considerably difficult, especially given the complicated nature of the sensors and actuators. The challenge stems from the closed tree structure of origami development, in which we arbitrarily select one point within each element that is surrounded by mountain and valley lines, then connect each of these points with a straight line. However, an open origami development structure allows for a much simpler origami robot. Although the actual gluing process did not go well during our robot’s demonstration, our entry still earned third prize in the contest [9].

Finally, I will end with a brief discussion about Japanese fans, which combine the characteristic functionality and beauty of Japanese manufacturing. People around the world have relied on fans to repel insects and start fires since time immemorial, but the folding fans were first created in Japan around 800 CE. These bellows-folded fans with bamboo bones behave differently than other iterations of single-sheet bellows folding. Famous painters of the Edo period (such as Katsushika Hokusai) attempted to create stories on fans by utilizing their various perspectives. But creating this type of composition is difficult, as each point on a 3D fan is distorted when the material is made flat for painting; for instance, a perfect two-dimensional circle on a fan painting becomes an ellipse once the fan is folded because the center of the fan painting’s arc differs from the pivot of the fan’s arc. Now that strain mathematics on fans are better understood [11], we can successfully utilize fan characteristics by creating characters and backgrounds that look different with a change of perspective (see Figure 3).

Figure 3. Active utilization of fan features creates images that appear different from changing viewpoints. Figure courtesy of Ichiro Hagiwara.

In conclusion, origami engineering has inspired a wide range of applications, from fashion statements to large-scale industrial structures. Due to remarkable progress in measurement technology, scientific and technological interests now span both micro and macro extremes without manufacturing equipment; instead, specialists rely on the self-folding function of origami structures. As such, origami engineering will become increasingly important in the coming years. I look forward to witnessing its enduring role in modern society.

Ichiro Hagiwara delivered an invited lecture on this topic at the 10th International Congress on Industrial and Applied Mathematics, which took place in Tokyo, Japan, last year.

[1] Diago, L., Shinoda, J., Yamazaki, K., & Hagiwara, I. (2023). Research on the maximization of the performance of arbitrary shaped kirigami honeycombs. Trans. JSME, 89(917) (in Japanese).
[2] Hagiwara, I. (2018). Birth of an “origami robot” with a new idea: Any three-dimensional object can be reproduced with origami. Asahi WebRonza. Retrieved from (in Japanese).
[3] Hagiwara, I., & Nadayosi, S. (2003). Folding process of cylindrical structures using origami model. Int. J. Automot. Eng., 34(4), 145-149 (in Japanese). 
[4] Hagiwara, I., Nara, C., & Yang, Y. (2022). Development of new foldable polyethylene terephthalate bottles. J. Adv. Simulat. Sci. Eng., 9(2), 247-262. 
[5] Ishida, S., Nojima, T., & Hagiwara, I. (2014). Mathematical approach to model foldable conical structures using conformal mapping. J. Mech. Des., 136(9), 091007.
[6] Nojima, T., & Hagiwara, I. (Eds.) (2012). Mathematics for origami and its application. Kyoritsu (in Japanese). 
[7] Nojima, T., & Saito, K. (2006). Development of newly designed ultra-light core structures. JSME Int. J. Ser. A Solid Mech. Mater. Eng., 49(1), 38-42.
[8] Origami engineering goes global: From PET bottles to space. (2007, December). Asahi Shimbun, 26 (in Japanese).
[9] Romero, J.A., Diago, L.A., Nara, C., Shinoda, J., & Hagiwara, I. (2016). Norigami folding machines for complex 3D shapes. In Proceedings of the ASME 2016 international design engineering technical conferences and computers and information in engineering conference. Volume 5B: 40th mechanisms and robotics conference. Charlotte, NC: ASME.
[10] Saito, K., Pellegrino, S., & Nojima, T. (2014). Manufacture of arbitrary cross-section honeycomb cores based on origami techniques. J. Mech. Des., 136(5), 051011.
[11] Yamazaki, K., Abe, F., & Hagiwara, I. (2021). Mathematical elucidation of the traditional Japanese fan focusing on its structure. Trans. JSME, 87(898) (in Japanese). 
[12] Zhao, X., Kong, C., Yang, Y., & Hagiwara, I. (2022). Reversed-torsion-type crush energy absorption structure and its inexpensive partial-heating torsion manufacturing method based on origami engineering. J. Manuf. Sci. Eng., 144(6), 061001.

Ichiro Hagiwara is a distinguished professor at Meiji University’s Research and Intellectual Property Strategy Organization, the Meiji Institute for Advanced Study of Mathematical Sciences, and the Meiji Institute of Autonomous Driving in Japan. He is an emeritus professor at the Tokyo Institute of Technology and holds a Ph.D. in mechanical engineering from the University of Tokyo. Hagiwara’s research interests pertain to origami engineering and intelligent self-driving vehicles.
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