# Father-Son Team Solves Geometry Problem With Infinite Folds

The result could help researchers answer a larger question about flattening objects from the fourth dimension to the third dimension.

## Introduction

The computer scientist Erik Demaine and his artist and computer scientist father, Martin Demaine, have been pushing the limits of paper folding for years. Their intricate origami sculptures are part of the permanent collection at the Museum of Modern Art, and a decade ago they were featured artists in a documentary about the art form that aired on PBS.

The pair started collaborating when Erik was 6 years old. “We had a company called the Erik and Dad Puzzle Company, which made and sold puzzles to toy stores across Canada,” said Erik Demaine, now a professor at the Massachusetts Institute of Technology.

Erik Demaine learned basic math and the visual arts from his father, but eventually taught Martin advanced math and computer science. “Now we’re both artists and both mathematicians/computer scientists,” Erik Demaine said. “We collaborate on many projects, especially those that span all of these disciplines.”

Their newest work, a mathematical proof, takes the collaboration to a new extreme: a realm where shapes collapse after being scored with infinitely many creases. It is an idea even they had a hard time accepting at first.

“We debated for a while, like, ‘Is this legit? Is this a real thing?’” said Erik Demaine, co-author of the new work along with Martin Demaine and Zachary Abel of MIT, Jin-ichi Itoh of Sugiyama Jogakuen University, Jason Ku of the National University of Singapore, Chie Nara of Meiji University and Jayson Lynch of the University of Waterloo.

The new work, posted online last May and published in the journal Computational Geometry in October, answers a question that the Demaines themselves posed in 2001 along with Erik’s doctoral adviser, Anna Lubiw of the University of Waterloo. They wanted to know whether it’s possible to take any polyhedral (or flat-sided) shape that’s finite (like a cube, rather than a sphere or the endless plane) and fold it flat using creases.

Cutting or tearing the shape isn’t allowed. Also, the shape’s intrinsic distances must be preserved. “This is just a fancy way to say, ‘You’re not allowed to stretch [or shrink] the material,’” Erik Demaine said. This type of folding must also avoid crossings, meaning “we don’t want the paper to pass through itself” because that doesn’t happen in the real world, he noted. Meeting this constraint is “especially challenging when everything’s moving continuously in 3D,” he added. Taken together, these constraints mean that simply squashing the shape won’t work.

The proof establishes that you can accomplish this folding, provided you resort to that infinite-creasing strategy, but it begins with a more down-to-earth technique that four of the same authors introduced in a 2015 paper.

There, they studied the folding question for a simpler class of shapes: orthogonal polyhedra whose faces meet at right angles and are perpendicular to at least one of the x, y and z coordinate axes. Meeting these conditions forces the faces of a shape to be rectangular, which makes the folding simpler, like collapsing a refrigerator box.

“That’s a relatively easy case to figure out, because each corner looks the same. It’s just two planes meeting perpendicularly,” Erik Demaine said.

After their 2015 success, the researchers set out to use their flattening technique to address all finite polyhedra. This change made the problem far more complex. This is because with non-orthogonal polyhedra, faces might have the shape of triangles or trapezoids — and the same creasing strategy that works for a refrigerator box won’t work for a pyramidal prism.

In particular, for non-orthogonal polyhedra, any finite number of creases always produces some creases that meet at the same vertex.

“That messed up our [folding] gadgets,” Erik Demaine said.

They considered different ways of circumventing this problem. Their explorations led them to a technique that’s illustrated when you try to flatten an object that is especially non-convex: a cube lattice, which is a kind of infinite grid in three dimensions. At each vertex in the cube lattice, many faces meet and share an edge, making it a formidable task to achieve flattening at any one of these spots.

“You wouldn’t necessarily think that you could, actually,” Ku said.

But considering how to flatten this type of notoriously challenging intersection led the researchers to the technique that ultimately powered the proof. First, they hunted for a spot “anywhere away from the vertex” that could be flattened, Ku said. Then they found another spot that could be flattened and kept repeating the process, moving closer to the problematic vertices and laying more of the shape flat as they moved along.

If they stopped at any point, they’d have more work to do, but they could prove that if the procedure went on forever, they could escape this issue.

“In the limit of taking smaller and smaller slices as you get to one of these problematic vertices, I will be able to flatten each one,” said Ku. In this context, the slices aren’t actual cuts but conceptual ones used to imagine breaking up the shape into smaller pieces and flattening it in sections, Erik Demaine said. “Then we conceptually ‘glue’ these solutions back together to obtain a solution to the original surface.”

The researchers applied this same approach to all non-orthogonal polyhedra. By moving from finite to infinite “conceptual” slices, they created a procedure that, taken to its mathematical extreme, produced the flattened object they were looking for. The result settles the question in a way that surprises other researchers who have engaged the problem.

“It just never even crossed my mind to use an infinite number of creases,” said Joseph O’Rourke, a computer scientist and mathematician at Smith College who has worked on the problem. “They changed the criteria of what constitutes a solution in a very clever way.”

For mathematicians, the new proof raises as many questions as it answers. For one, they’d still like to know whether it’s possible to flatten polyhedra with only finitely many creases. Erik Demaine thinks so, but his optimism is based on a hunch.

“I’ve always felt like it should be possible,” he said.

The result is an interesting curiosity, but it could have broader implications for other geometry problems. For instance, Erik Demaine is interested in trying to apply his team’s infinite-folding method to more abstract shapes. O’Rourke recently suggested that the team investigate whether they could use it to flatten four-dimensional objects down to three dimensions. It’s an idea that might have seem far-fetched even a few years ago, but infinite folding has already produced one surprising result. Maybe it can generate another.

“The same type of approach might work,” said Erik Demaine. “It’s definitely a direction to explore.”