A father-son team solves a geometry problem with infinite folds

After their success in 2015, the researchers decided to use their flattening technique to process all finite polyhedra. This change made the problem much more complex. Indeed, with non-orthogonal polyhedra, the faces can be shaped like triangles or trapezoids – and the same folding strategy that works for a refrigerator box won’t work for a pyramid prism.

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

“It ruined our [folding] gadgets,” said Erik Demaine.

They considered different ways to circumvent this problem. Their explorations led them to a technique that shines when trying to flatten a particularly non-convex object: a cubic lattice, which is a sort of infinite three-dimensional grid. At each vertex of the cubic lattice, many faces meet and share an edge, making it a formidable task to achieve the flattening at any of these points.

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

But thinking about how to flatten this notoriously difficult type of intersection led researchers to the technique that ultimately fueled the evidence. First, they looked for a spot “anywhere off the top” that could be flattened, Ku said. Then they found another spot that could be flattened and kept repeating the process, getting closer to the problematic peaks and flattening the shape more as they went.

If they stopped at some point, they would have more work to do, but they could prove that if the procedure lasted indefinitely, they could escape this problem.

“Within the limit of taking smaller and smaller slices as you get to one of those problematic peaks, I’ll be able to flatten each one out,” Ku said. In this context, slices are not actual cuts, but conceptual cuts used to imagine dividing the shape into smaller pieces and flattening it into sections, said Erik Demaine. “Then we conceptually ‘glue’ those solutions together to get a solution at the original surface.”

The researchers applied this same approach to all non-orthogonal polyhedra. Going from finite to infinite “conceptual” slices, they created a procedure which, 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 looked into the problem.

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

For mathematicians, the new evidence raises as many questions as it answers. On the one hand, they would still like to know if it is possible to flatten polyhedra with only a finite number of folds. Erik Demaine thinks so, but his optimism is based on an intuition.

“I always thought it should be possible,” he said.

The result is an interesting curiosity, but it could have wider implications for other geometry problems. For example, Erik Demaine wants to try applying his team’s infinity folding method to more abstract shapes. O’Rourke recently suggested the team investigate whether they could use it to flatten four-dimensional objects into three dimensions. It’s an idea that might have seemed far-fetched just a few years ago, but infinite folding has already produced a surprising result. Maybe that can generate another one.

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

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

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