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Folding a New Tomorrow: Origami Meets Math and Science

MAA Distinguished Lecture: Origami Ain’t What It Used to Be

Paper folding has come a long way. You’ve creased a square into a crane, perhaps, but can you similarly transform a 1’ by 10’ rectangle into a cuckoo clock? Or replicate, using only straight folds, the sinuous curves of a rose blossom or the plated abdomen of a wasp?

Speaking at the MAA Carriage House on November 10, Thomas Hull (Western New England University) not only wowed listeners with such wondrous feats of folding, but also argued that origami intersects with mathematical subfields from matrix algebra and graph theory to statistical mechanics and topology. And then there’s the practical side.

“Over the past five years or so there’s been a big interest from mathematicians to engineers to physicists to really look at origami as inspiration in applications,” Hull said.

Titled “Folding a New Tomorrow: Origami Meets Math and Science,” Hull’s talk was part of MAA’s NSA-funded Distinguished Lecture Series.

Try This

Origami is the stuff of theorems these days, and Hull led audience members to discover one result—Maekawa’s Theorem—for themselves. 

Take a square of paper and draw a point in the center. Fold the paper such that the crease goes through the point, leaving the paper folded. Fold through the point a few more times, then unfold the paper and count the mountain (^) and valley (v) folds emanating from the point. Do this with three or four pieces of paper, and record your results. 

Notice anything? How do the number of mountain folds and the number of valley folds relate to each other?

Once you’ve hypothesized a relationship, try proving it. Hull deployed what he called “the classic proof by monorail.” To get started down that route, refold one of your papers and cut off the vertex, revealing a polygonal cross section. 

Now imagine a monorail traveling clockwise around the cross section. How much does the train rotate when it encounters a mountain fold? A valley fold? What is the train’s total rotation when it reaches its starting point? Work it out.

How to Fold a Map

It’s not all imaginary monorails and easy algebra, of course. Mathematicians use rotation matrices to model flat pieces of paper folding into three-dimensional shapes, and such matrix models of rigid origami enable computer visualizations. Check out, for instance, the Miura map fold.     

Japanese astrophysicist Koryo Miura devised this fold to solve the problem of outfitting satellites with large solar panels. Folded Miura’s way, a panel can collapse small enough to fit inside a capsule and then unfold—via an expansion rod, say—in outer space.

“It won’t catch on itself,” Hull said. “As opposed to the solar panels on the International Space Station, which are just accordion pleated.”

During an attempted collapse in the early 2000s, the panels buckled.

“They had to get a guy to space walk and actually do origami in outer space to push all the mountains and valleys in the right direction again,” Hull remembered. “If only they used the Miura map fold!”

Its utility aside, natural questions about the Miura map fold turn up some surprising connections. The number of mountain-valley assignments that will allow an m by n Miura fold to lie flat equals the number of ways to properly 3-vertex color an m by n grid graph (with one vertex pre-colored), for instance. 

Small-scale 

One refrain of Hull’s talk was the ability of advances in origami-related mathematics to thrill physicists and engineers—and inspire them. 

Consider materials physicists Itai Cohen (Cornell University), Ryan Hayward, and Chris Santangelo (both University of Massachusetts). Theoretical mathematical results have informed their development of self-folding polymer gels.

“Have you ever played with grow animals?” Hull asked listeners. “You stick them in water; they grow.”

The physicists took some “grow animal stuff”—also known as hydrogel—and sandwiched it between layers of a rigid polystyrene-type plastic, Hull explained. Then they carved crease lines into the plastic, creating troughs of exposed hydrogel.

When the scored sandwich is thrown in water, the hydrogel swells along the crease lines. Troughs on the top and bottom force mountain and valley folds, respectively. The angle of folding varies with the width of the trough.

With an NSF grant to explore and develop this self-folding process, Hull and the materials physicists succeeded in getting a polymer gel to collapse—by itself—via the Miura map fold. Between the small—less than a millimeter—scale and the bio-compliant material, the team’s creation—or something like it, anyway—could find its way into the human body someday.  

“For applications that people have been dreaming about using origami for biomechanical work,” Hull said, “this could be useful.”

Hull is happy to see interest in origami extend beyond hobbyists and mathematicians, but the upswell of academic attention does necessitate curation. Researchers too often “reinvent the wheel” because they’re unaware of past work, Hull explained, and not everyone adopts the same notation as they churn out new results.

“I think one of the challenges is to try to collect all this and make it more uniform so that it’s easier for people to get on the bandwagon and learn this stuff,” Hull said.

Katharine Merow is a freelance writer living in Washington, D.C.