FALCONIUM

Creating an animal from a single uncut piece of paper is one of life's greatest simple pleasures. Few other activities are as rewarding in relation to their brevity and convenience as origami, with the classic crane holding the distinction of being the most widely recognized and adored creation of this art form. Yet as satisfying as folding thousands of cranes, frogs, and boats for decorative purposes is, origami has also recently made its way into engineering, proving itself to be extremely practical and astonishingly useful as well as aesthetically pleasing.

Origami owes this new surge in appreciation and application to its inherent mathematics. Physicist and engineer Robert J. Lang, who specializes in full-time paper folding and is considered one of the world's greatest origami designers, notes that mathematics makes it physically possible to fold nearly anything out of a sheet of paper, from scorpions with spindly appendages to violinists.


Tom Hull, mathematician at Merrimack College, corroborates this relationship, calling origami “latent mathematics”. Origami often makes use of geometric patterns, allowing for numerous polyhedral possibilities. For example, computer scientists at the University of Waterloo and SUNY at Stony Brook proved in 1999 that theoretically, any complex polyhedron of two colors can be created by folding a two-colored piece of paper (the origami checkerboard is physical evidence of this theorem). Origami is also a question of optimization: how to create an object with the minimal amount of paper used.


Lang developed a computer algorithm, TreeMaker, which outputs the crease pattern for a complex origami design when a stick figure of the subject's essential features is inputted. TreeMaker generates this base by treating the process as a nonlinear constrained optimization problem, converting the stick figure into a series of algebraic equations, and then using a special numerical optimization code, CFSQP, to solve the equations. Essentially, the algorithm finds a local maximum and expresses it as a pattern of folds. TreeMaker can compute the crease pattern for more complex shapes than can be done by hand, and minimizes paper used while doing so. (Though TreeMaker cannot yet compute crease assignments or the sequence of folds, these can be determined by hand.) Origami's innate intimacy with math gives it relative simplicity and convenience, easing its application to engineering.


One of origami's most ambitious engineering applications is the current quest to develop a super-sized telescope forty times stronger than the Hubble Space Telescope, which is currently the largest one in space with an aperture of 2.4 meters. In 2002, scientists at the Lawrence Livermore National Laboratory (LLNL) created a prototype of the proposed gargantuan telescope with a lens diameter of 5 meters. The actual telescope will be a transmissive telescope, with the thin plastic, 100-meter diameter diffractive main lens a few kilometers away from the accessory lens in space. Its immense size will allow it to absorb more light, granting it the power to view terrestrial footballs in one direction and celestial bodies light-years away, such as extra-solar system planets, in the other direction. Initially, however, there was a significant obstacle to this goal. The telescope must be shipped up to space, but no shuttles have room for a fully expanded 100-meter lens. All current shuttles have a maximum cylindrical space four meters in diameter and ten meters long. Thus, the telescope must be carefully folded before it can be stored, and due to the fragility of the lens and the irreparable damage unplanned folds can inflict, it requires a collapsing pattern that minimizes the number of predetermined creases and allows it to fit within a cylinder. To complicate matters further, the folding pattern must avoid sharp creases to avoid impairing optical performance. With the help of Lang and origami principles, a successful solution, resembling an umbrella shape, was found.


Other technological applications abound for origami. Origami has been used in airbag design and simulation (the best way to fold an airbag to allow for effective deployment and operation was found using Lang's insect algorithm); making strong aluminum cans; crushable plastic bottles; conveniently foldable maps; foldable satellite antennas (a technique called Miura-ori allows satellite antennas to be easily folded, then unfolded in space); heat shielding; collision safety engineering; solar sails (mechanism that launches shuttles out of the solar system by riding on the particles in solar wind); environmentally-friendly and exceptionally sturdy pots; planes of sugarcane fiber that can be launched from the ISS; and stints that can be inserted in their folded form into the abdominal aorta, then expand to support a damaged artery, and allow boxes made of DNA to deliver drugs to diseased cells.


The airbag is a particularly useful application. Airbag designers know that lives depend on the quality and effectiveness of airbags, which must wholly expand in milliseconds and provide firm support as well as cushioning to prevent injury. Lang has developed virtual folding patterns for airbags to allow simulations to be conducted before an airbag is installed in a car. Simulations are very difficult, involving complex techniques such as finite element analysis to create virtual scenarios. Finite element analysis divides airbags into triangles and positions the triangles as the bag inflates according to factors such as elasticity and shape of the bag. Using an algorithm known as the universal molecule, the simulation airbag can be flattened with the polygons adjacent to each other. Clearly, the principles of origami are crucial in evaluating the effectiveness and safety of a proposed airbag design before it is and put into use. Thus, these simulations have financial benefits in addition to ultimately minimizing injuries and deaths in car accidents.


The possibilities of origami are endless. Though once considered a mere artistic hobby, the Japanese art form has now propelled itself into the realms of science and engineering, and may one day allow us to receive clear images of distant planets or save us through an inflating airbag, deftly folded to maximize efficacy. By digging deeper into the mathematics of origami and combining folding principles with engineering needs, scientists continue to innovate and invent, developing technologies that can shape our world and our understanding of it. Clearly, the principles of uncut and folded sheets of paper will have enormous implications on the future of society and science.

Website to check out:
www.langorigami.com

-Ling Jing

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Cipra, Barry A. "In the Fold: Origami Meets Mathematics." SIAM News. Web. 31 Oct 2009.

"Extreme Origami: Fold Everything." National Geographic Oct. 2009: 24-27. Print.
Robert J. Lang Origami. Web. 31 Oct 2009.

"The Science Of Origami." Web Japan. 12 Mar. 2008. Web. 31 Oct. 2009.