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After their 2015 success, the researchers got down to use their flattening method to deal with all finite polyhedra. This modification made the issue way more complicated. It’s because with non-orthogonal polyhedra, faces might need the form of triangles or trapezoids—and the identical creasing technique that works for a fridge field received’t work for a pyramidal prism.
Specifically, for non-orthogonal polyhedra, any finite variety of creases at all times produces some creases that meet on the similar vertex.
“That tousled our [folding] devices,” Erik Demaine stated.
They thought-about other ways of circumventing this drawback. Their explorations led them to a way that’s illustrated while you attempt to flatten an object that’s particularly non-convex: a dice lattice, which is a sort of infinite grid in three dimensions. At every vertex within the dice lattice, many faces meet and share an edge, making it a formidable job to attain flattening at any certainly one of these spots.
“You wouldn’t essentially suppose that you might, really,” Ku stated.
However contemplating methods to flatten this sort of notoriously difficult intersection led the researchers to the method that finally powered the proof. First, they hunted for a spot “anyplace away from the vertex” that might be flattened, Ku stated. Then they discovered one other spot that might be flattened and stored repeating the method, shifting nearer to the problematic vertices and laying extra of the form flat as they moved alongside.
In the event that they stopped at any level, they’d have extra work to do, however they may show that if the process went on eternally, they may escape this subject.
“Within the restrict of taking smaller and smaller slices as you get to certainly one of these problematic vertices, I can flatten each,” stated Ku. On this context, the slices aren’t precise cuts however conceptual ones used to think about breaking apart the form into smaller items and flattening it in sections, Erik Demaine stated. “Then we conceptually ‘glue’ these options again collectively to acquire an answer to the unique floor.”
The researchers utilized this similar method to all non-orthogonal polyhedra. By shifting from finite to infinite “conceptual” slices, they created a process that, taken to its mathematical excessive, produced the flattened object they have been in search of. The outcome settles the query in a approach that surprises different researchers who’ve engaged the issue.
“It simply by no means even crossed my thoughts to make use of an infinite variety of creases,” stated Joseph O’Rourke, a pc scientist and mathematician at Smith Faculty who has labored on the issue. “They modified the factors of what constitutes an answer in a really intelligent approach.”
For mathematicians, the brand new proof raises as many questions because it solutions. For one, they’d nonetheless wish to know whether or not it’s potential to flatten polyhedra with solely finitely many creases. Erik Demaine thinks so, however his optimism relies on a hunch.
“I’ve at all times felt prefer it ought to be potential,” he stated.
The result’s an fascinating curiosity, however it might have broader implications for different geometry issues. As an example, Erik Demaine is concerned about attempting to use his crew’s infinite-folding technique to extra summary shapes. O’Rourke just lately instructed that the crew examine whether or not they might use it to flatten four-dimensional objects down to 3 dimensions. It’s an thought that may have appear far-fetched even a couple of years in the past, however infinite folding has already produced one stunning outcome. Possibly it might probably generate one other.
“The identical sort of method may work,” stated Erik Demaine. “It’s positively a course to discover.”
Authentic story reprinted with permission from Quanta Journal, an editorially impartial publication of the Simons Basis whose mission is to boost public understanding of science by overlaying analysis developments and developments in arithmetic and the bodily and life sciences.
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