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.”
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.