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Origami Research Papers

Origami Research and Applications

The uses of origami are not confined to artistic expression. Insights from folding paper have been applied to space technology, automobiles, medicine and programmable matter; these applications began to appear a few decades ago. At heart is the mathematics behind the humble piece of paper, which was studied as early as the 1930s. The mathematics gives rise to challenging unsolved problems and software that design complex origami on-demand.

This article lists some of the more accessible research as it is meant as a primer to origami research, but it may be somewhat outdated.

Origami in Technology

Solar Panels for a Satellite (1995)

Airbag Simulations

Origami Stent Graft (2003)

Closed and open forms of the stent graft
Researchers from the University of Oxford developed an origami stent graft, a tube-shaped medical device used to hold arteries open during surgery. The metal tube would be folded into a thinner shape using a special origami pattern. It would then be inserted into an artery, where body heat would cause it to unfold and expand, holding the artery open.

Image source: http://www-civil.eng.ox.ac.uk/people/zy/old/research/stent.pdf


Other Technologies

Structural Origami

Deployable Structures: Structures that have different initial and final configurations, folding from one to the other

Automated Origami Design

Programmers have written software that allow users to specify a desired origami shape and calculate its blueprint (crease pattern) based on that data. Varying programs have been written to design different types of origami, from animals to geometric shapes.

Treemaker (1998)

Robert J. Lang developed the mathematics needed to generate a pattern of lines that folds into a specified stick figure, automating the design of animal origami.

Origamizer (2008)

Other Automated Origami Design Algorithms

Origami Mathematics

Creases on paper form lines. Different creases meet in point and form angles. Naturally, origami is deeply tied to geometry. Seeing an origami model as a mathematical object obeying certain mathematical “laws of folding” is at the heart of every computer program used in Automated Origami Design. Problems in mathematics itself have even been solved using origami techniques.

Flat-foldable Origami

Crease pattern of a flat-folded bird
Much of origami is folded into a flat product before it is shaped into a more lifelike 3D form. Observations regarding the blueprints of flat origami formed the early “laws of folding”. One such “law” is that the regions that the blueprint divides the paper into can be colored in two colors so that no neighboring regions have the same color.

Image source: Robert J. Lang. The math and magic of origami. TED.com (2008).


Fold-and-Cut Problem

Fold-and-cut swan

Other Mathematics in Origami

OSME Conferences

Worldwide origami research
Many of the above pieces of research were presented at the International Conference on Origami in Science, Mathematics and Education (OSME), which inspired researchers and provided a platform for them to discuss and learn of the newest developments.

In 1989, origami researchers gathered in Ferrara, Italy for the First International Meeting of Origami Science and Technology (1OSME), where they discussed their research findings and explored new directions. This has since evolved into a highly successful series of OSME conferences. They link separate research groups from around the world, forming an international community for origami research.

Educational uses of origami have been a focus since 3OSME, the third conference in 2001. This acknowledges the possibilities of origami in demonstrating abstract geometric concepts through fun and engaging hands-on activities.


Further Resources

Many more examples of origami research and applications can be found in:
  • OSME conference proceedings:  3OSME, 4OSME, 5OSME
  • The conference programs have links to the abstracts: 4OSME, 5OSME
Technologies that apply origami are springing up rapidly these years, with many expected to appear in the 6OSME.

Some videos are excellent introductions to origami:

Some books go into more detail about origami mathematics and computer science:
In 1955, Japanese scientists used an origami technique, the Miura map fold, to fold the huge solar panels of a satellite into a small package. After the satellite entered space, the solar panels unfolded to full size and functionality.
Opening the Miura map fold
Artist's impression of a satellite with the Miura map fold
Stick figure representing the subject
Crease pattern ("folding instructions")
User-specified 3D shape, and the computed crease pattern

In a 1999 paper, Erik Demaine — now an MIT professor of electrical engineering and computer science, but then an 18-year-old PhD student at the University of Waterloo, in Canada — described an algorithm that could determine how to fold a piece of paper into any conceivable 3-D shape.

It was a milestone paper in the field of computational origami, but the algorithm didn’t yield very practical folding patterns. Essentially, it took a very long strip of paper and wound it into the desired shape. The resulting structures tended to have lots of seams where the strip doubled back on itself, so they weren’t very sturdy.

At the Symposium on Computational Geometry in July, Demaine and Tomohiro Tachi of the University of Tokyo will announce the completion of a quest that began with that 1999 paper: a universal algorithm for folding origami shapes that guarantees a minimum number of seams.

“In 1999, we proved that you could fold any polyhedron, but the way that we showed how to do it was very inefficient,” Demaine says. “It’s efficient if your initial piece of paper is super-long and skinny. But if you were going to start with a square piece of paper, then that old method would basically fold the square paper down to a thin strip, wasting almost all the material. The new result promises to be much more efficient. It’s a totally different strategy for thinking about how to make a polyhedron.”

Demaine and Tachi are also working to implement the algorithm in a new version of Origamizer, the free software for generating origami crease patterns whose first version Tachi released in 2008.

Maintaining boundaries

The researchers’ algorithm designs crease patterns for producing any polyhedron — that is, a 3-D surface made up of many flat facets. Computer graphics software, for instance, models 3-D objects as polyhedra consisting of many tiny triangles. “Any curved shape you could approximate with lots of little flat sides,” Demaine explains.

Technically speaking, the guarantee that the folding will involve the minimum number of seams means that it preserves the “boundaries” of the original piece of paper. Suppose, for instance, that you have a circular piece of paper and want to fold it into a cup. Leaving a smaller circle at the center of the piece of paper flat, you could bunch the sides together in a pleated pattern; in fact, some water-cooler cups are manufactured on this exact design.

In this case, the boundary of the cup — its rim — is the same as that of the unfolded circle — its outer edge. The same would not be true with the folding produced by Demaine and his colleagues’ earlier algorithm. There, the cup would consist of a thin strip of paper wrapped round and round in a coil — and it probably wouldn’t hold water.

“The new algorithm is supposed to give you much better, more practical foldings,” Demaine says. “We don’t know how to quantify that mathematically, exactly, other than it seems to work much better in practice. But we do have one mathematical property that nicely distinguishes the two methods. The new method keeps the boundary of the original piece of paper on the boundary of the surface you’re trying to make. We call this watertightness.”

A closed surface — such as a sphere — doesn’t have a boundary, so an origami approximation of it will require a seam where boundaries meet. But “the user gets to choose where to put that boundary,” Demaine says. “You can’t get an entire closed surface to be watertight, because the boundary has to be somewhere, but you get to choose where that is.”

Lighting fires

The algorithm begins by mapping the facets of the target polyhedron onto a flat surface. But whereas the facets will be touching when the folding is complete, they can be quite far apart from each other on the flat surface. “You fold away all the extra material and bring together the faces of the polyhedron,” Demaine says.

Folding away the extra material can be a very complex process. Folds that draw together multiple faces could involve dozens or even hundreds of separate creases.

Developing a method for automatically calculating those crease patterns involved a number of different insights, but a central one was that they could be approximated by something called a Voronoi diagram. To understand this concept, imagine a grassy plain. A number of fires are set on it simultaneously, and they all spread in all directions at the same rate. The Voronoi diagram — named after the 19th-century Ukrainian mathematician Gyorgy Voronoi — describes both the location at which the fires are set and the boundaries at which adjacent fires meet. In Demaine and Tachi’s algorithm, the boundaries of a Voronoi diagram define the creases in the paper.

“We have to tweak it a little bit in our setting,” Demaine says. “We also imagine simultaneously lighting a fire on the entire polygon of the polyhedron and growing out from there. But that concept was really useful. The challenge is to set up where to light the fires, essentially, so that the Voronoi diagram has all the properties we need.”

Completed quest

“It’s very impressive stuff,” says Robert Lang, one of the pioneers of computational origami and a fellow of the American Mathematical Society, who in 2001 abandoned a successful career in optical engineering to become a full-time origamist. “It completes what I would characterize as a quest that began some 20-plus years ago: a computational method for efficiently folding any specified shape from a sheet of paper. Along the way, there have been several nice demonstrations of pieces of the puzzle: an algorithm to fold any shape, but not very efficiently; an algorithm to efficiently fold particular families of tree-like shapes, but not surfaces; an algorithm to fold trees and surfaces, but not every shape. This one covers it all! The algorithm is surprisingly complex, but that arises because it is comprehensive. It truly covers every possibility. And it is not just an abstract proof; it is readily computationally implementable.”

Joseph O’Rourke, a professor of mathematics and computer science at Smith College and the author of How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra, agrees. “What was known before was either ‘cheating’ — winding the polyhedron with a thin strip — or not guaranteed to succeed,” he says. “Their new algorithm is guaranteed to produce a folding, and it is the opposite of cheating in that every facet of the polyhedron is covered by a ‘seamless’ facet of the paper, and the boundary of the paper maps to the boundary of the polyhedral manifold — their ‘watertight’ property. Finally, the extra structural ‘flash’ needed to achieve their folding can all be hidden on the inside and so is invisible.”