After their 2015 success, the researchers got down to use their flattening approach to handle all finite polyhedra. This alteration made the issue way more complicated. It is because with non-orthogonal polyhedra, faces may need the form of triangles or trapezoids—and the identical creasing technique that works for a fridge field gained’t work for a pyramidal prism.
Particularly, for non-orthogonal polyhedra, any finite variety of creases all the time produces some creases that meet on the similar vertex.
“That tousled our [folding] devices,” Erik Demaine stated.
They thought of other ways of circumventing this downside. Their explorations led them to a way that’s illustrated whenever you attempt to flatten an object that’s particularly non-convex: a dice lattice, which is a form 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 realize flattening at any considered one of these spots.
“You wouldn’t essentially assume that you might, really,” Ku stated.
However contemplating the best way to flatten such a notoriously difficult intersection led the researchers to the approach that finally powered the proof. First, they hunted for a spot “anyplace away from the vertex” that may very well be flattened, Ku stated. Then they discovered one other spot that may very well be flattened and saved 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 might show that if the process went on ceaselessly, they might escape this challenge.
“Within the restrict of taking smaller and smaller slices as you get to considered one of these problematic vertices, I can flatten every one,” 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 had been searching for. The outcome settles the query in a method 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 standards of what constitutes an answer in a really intelligent method.”
For mathematicians, the brand new proof raises as many questions because it solutions. For one, they’d nonetheless prefer to know whether or not it’s attainable to flatten polyhedra with solely finitely many creases. Erik Demaine thinks so, however his optimism is predicated on a hunch.
“I’ve all the time felt prefer it needs to be attainable,” he stated.
The result’s an attention-grabbing curiosity, however it might have broader implications for different geometry issues. As an illustration, Erik Demaine is serious about attempting to use his staff’s infinite-folding technique to extra summary shapes. O’Rourke just lately instructed that the staff examine whether or not they might use it to flatten four-dimensional objects down to 3 dimensions. It’s an concept which may have appear far-fetched even a number 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 would possibly work,” stated Erik Demaine. “It’s positively a path to discover.”
Authentic story reprinted with permission from Quanta Journal, an editorially impartial publication of the Simons Basis whose mission is to reinforce public understanding of science by masking analysis developments and traits in arithmetic and the bodily and life sciences.