Following their success in 2015, the researchers started using their flattening technique for all finite polyhedra. This change complicates matters. This is because for a non-orthogonal polyhedron, the faces may have the shape of a triangle or a trapezoid – the same crease strategy that works for a refrigerator box doesn’t work for a pyramid.
In particular, for non-orthogonal polyhedra, any finite number of folds will always produce a few folds that intersect at the same vertex.
“It messed up our [folding] gadgets,” Erik Demaine said.
They considered different ways to circumvent this problem. Their exploration led them to a technique that is illustrated when you try to flatten a particularly non-convex object: a cubic lattice, which is an infinite grid in three dimensions. At each vertex of a cubic lattice, many faces intersect and share an edge, making it a difficult task to achieve flattening at any of these points.
“Actually, you don’t necessarily think you can,” Ku said.
But given how to flatten this notoriously challenging intersection, the researchers found the technique that ultimately powers the proof. First, they looked for “anywhere far from the vertex” where they could flatten out, Ku said. They then found another point that could be flattened and kept repeating the process, getting closer to the vertices in question and flattening more shapes as they moved.
If they stop at any point, they’ll have more work to do, but they can prove they can get away with the problem if the process goes on forever.
“When you get to one of these problematic vertices, within the constraints of taking smaller and smaller slices, I’ll be able to flatten each slice,” Ku said. In this case, slicing isn’t an actual cut, but a conceptual slice that imagines breaking down a shape into smaller pieces and flattening it, Erik Demaine says. “We then conceptually ‘glue’ these solutions together to obtain a solution for the original surface.”
The researchers applied the same method to all non-orthogonal polyhedra. By moving from a finite slice of “concepts” to an infinite slice of “concepts”, they created a program that took the math to an extreme, producing the flat objects they were looking for. The results solved the problem in a way that surprised other researchers working on the problem.
“I never thought to use an infinite number of creases,” says Joseph O’Rourke, a computer scientist and mathematician at Smith College, who has studied the problem. “They’ve changed the criteria for what constitutes a solution in a very clever way.”
For mathematicians, the new proof raises as many questions as it answers. On the one hand, they still wondered if it was possible to flatten a polyhedron with limited creases. Erik Demaine thinks so, but his optimism is based on a hunch.
“I’ve always felt that it should be possible,” he said.
The result is an interesting curiosity, but one that could have wider implications for other geometric problems. For example, 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 a four-dimensional object into three dimensions. The idea might have seemed far-fetched a few years ago, but infinite folding has yielded a surprising result. Maybe it can spawn another.
“The same approach might work,” says Erik Demaine. “It’s definitely a direction to explore.”
ability Reprinted with permission Quanta Magazine, Editing independent publications Simmons Foundation Its mission is to enhance public understanding of science by covering research developments and trends in mathematics as well as the physical and life sciences.