Noperthedron Discovery Resolves Centuries-Old Geometric Mystery

Key Quotes

"One of the lovely things about math is that it's full of these problems that are simple to state but conceal great puzzling depth. Take the famous Collatz conjecture. You can start with any positive integer: 10, or 71, or 1,635,332. If it's even, you divide it by two. If it's odd, you multiply it by three and add one. You do that over and over again. For every number that has ever been tested, up to extremely big ones, it always eventually ends up at one. Try it yourself, it'll work. But that doesn't make it mathematically true. No proof exists that shows that this is true for all possible integers, and it would take some kind of great mathematical innovation for that to happen."

This quote highlights the nature of certain mathematical problems, using the Collatz conjecture as an example. Klarreich explains that these problems are characterized by their simple statements that hide complex challenges, and that empirical testing, while suggestive, does not constitute mathematical proof.

"The history of this problem dates back to this really fascinating person named Prince Rupert of the Rhine, who had this incredibly colorful life. He was the son of a German prince, but like most royalty of that time, he was also related to many other royal houses. Among other things, he was the nephew of King Charles of England. Prince Rupert fought for Charles in the English Civil War and rose to great prominence. In fact, even though he was only in his twenties, he rose to be the commander of the armed forces in the English Civil War."

Klarreich introduces Prince Rupert of the Rhine, detailing his multifaceted life as a prince, military commander, privateer, and governor. This historical context sets the stage for the geometric problem that bears his name, emphasizing his adventurous and prominent career.

"So we're talking about geometric objects that are called convex polyhedra, which is a bit of a mouthful. But these are actually very familiar shapes in the world. These are things that have flat faces and no holes inside them or weird protuberances. So think of something like a cube or a pyramid or a soccer ball, except not rounded. It should have flat faces. Those are examples of convex polyhedra."

Klarreich defines the term "convex polyhedra" by relating it to familiar shapes. She explains that these are objects with flat faces and no internal or external irregularities, providing examples like cubes and pyramids to make the concept accessible.

"So what's really new in the past decade or so in this problem is that you can start using computers to try to tackle shapes that you really couldn't tackle with pencil and paper. So mathematicians tried using computers to analyze all kinds of shapes, hundreds of millions of shapes. For almost every single one of those, the computer found a Rupert tunnel, no problem, instantly. But there were a few shapes that the computer didn't find a tunnel for, even after running for quite a while."

Klarreich discusses the impact of computers on solving this geometric problem. She explains that computational analysis has allowed mathematicians to test a vast number of shapes, finding Rupert tunnels for most, but also identifying a few shapes for which tunnels could not be found, suggesting potential counterexamples.

"So they specifically constructed this shape to try to make something that would not have a Rupert tunnel. Okay. If you have a shape and you suspect or hope that it's a Nopert, then what you need to do is you need to somehow show that no matter how you orient the two shapes, you cannot pass through."

Klarreich describes the intentional design of the Noperthedron by its creators. She explains that the mathematicians constructed this specific shape with the goal of demonstrating that it could not pass through an identical copy of itself, thus proving it to be a "Nopert."

"So then the question is, well, how do you do that when you have infinitely many different ways to orient the shapes? That's why a computer can't do it. But you do have something that helps you. So suppose you choose some pair of orientations for the shape and you find that the shadow of the one you're trying to pass through is too big. It doesn't fit through, like that shadow sticks out in some places, right? So you've ruled out one orientation, but you've actually done more than that because if you make a very small change to the position of that shape, then there's just going to be a very small change to its shadow."

Klarreich addresses the challenge of proving the absence of a Rupert tunnel when there are infinite possible orientations. She explains that by analyzing the "shadows" cast by the shapes and observing how small changes affect these shadows, mathematicians can rule out not just single orientations but entire families of them.