# Algorithm generates origami folding patterns for any shape

June 23, 2017

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."
-end-

Paper: "Origamizer: A Practical Algorithm for Folding Any Polyhedron http://erikdemaine.org/papers/Origamizer_SoCG2017/paper.pdf

ARCHIVE: "Super Mario Brothers" is hard http://news.mit.edu/2016/mario-brothers-hard-complexity-class-pspace-0601

ARCHIVE: Origami robot folds itself up, crawls away http://news.mit.edu/2014/mobile-folding-robots-0807

ARCHIVE: Bake your own robot http://news.mit.edu/2014/bake-your-own-robot-0530

ARCHIVE: The math of the Rubik's cube http://news.mit.edu/2011/rubiks-cube-0629

Massachusetts Institute of Technology

## Related Algorithm Articles from Brightsurf:

CCNY & partners in quantum algorithm breakthrough
Researchers led by City College of New York physicist Pouyan Ghaemi report the development of a quantum algorithm with the potential to study a class of many-electron quantums system using quantum computers.

Machine learning algorithm could provide Soldiers feedback
A new machine learning algorithm, developed with Army funding, can isolate patterns in brain signals that relate to a specific behavior and then decode it, potentially providing Soldiers with behavioral-based feedback.

New algorithm predicts likelihood of acute kidney injury
In a recent study, a new algorithm outperformed the standard method for predicting which hospitalized patients will develop acute kidney injury.

New algorithm could unleash the power of quantum computers
A new algorithm that fast forwards simulations could bring greater use ability to current and near-term quantum computers, opening the way for applications to run past strict time limits that hamper many quantum calculations.

QUT algorithm could quash Twitter abuse of women
Online abuse targeting women, including threats of harm or sexual violence, has proliferated across all social media platforms but QUT researchers have developed a sophisticated statistical model to identify misogynistic content and help drum it out of the Twittersphere.

New learning algorithm should significantly expand the possible applications of AI
The e-prop learning method developed at Graz University of Technology forms the basis for drastically more energy-efficient hardware implementations of Artificial Intelligence.

Algorithm predicts risk for PTSD after traumatic injury
With high precision, a new algorithm predicts which patients treated for traumatic injuries in the emergency department will later develop posttraumatic stress disorder.

New algorithm uses artificial intelligence to help manage type 1 diabetes
Researchers and physicians at Oregon Health & Science University have designed a method to help people with type 1 diabetes better manage their glucose levels.

A new algorithm predicts the difficulty in fighting fire
The tool completes previous studies with new variables and could improve the ability to respond to forest fires.

New algorithm predicts optimal materials among all possible compounds
Skoltech researchers have offered a solution to the problem of searching for materials with required properties among all possible combinations of chemical elements.

Read More: Algorithm News and Algorithm Current Events
Brightsurf.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com.