First shape found that can't pass through itself

Summary

But such resistance doesn’t prove that a shape is a Nopert. There are infinitely many ways to orient a shape, and a computer can only check finitely many. Researchers don’t know whether the holdouts are true Noperts or just shapes whose Rupert passages are hard to find. What they do know is that candidate Noperts are incredibly rare. Starting last year, Murphy began to construct hundreds of millions of shapes. These include random polyhedra, polyhedra whose vertices lie on a sphere, polyhedra with special symmetries, and polyhedra in which he moved one vertex to intentionally mess up a previous Rupert passage. His algorithm easily found Rupert tunnels for nearly every one. The contrast between these quick results and the stubbornness of the Nopert holdouts made some mathematicians suspect that true Noperts do exist. But until August, all they had were suspicions. No Passage Steininger, now 30, and Yurkevich, 29, have been friends since they participated together as teenagers in mathematics Olympiad competitions. Even though both eventually left academia (after a doctorate for Yurkevich and a master’s for Steininger), they have continued to explore unsolved problems together. “We just had pizza three hours ago, and we talked about math almost the whole time,” Steininger told Quanta. “That’s what we do.” Five years ago, the pair happened upon a YouTube video of one cube passing through another, and they were instantly smitten. They developed an algorithm to search for Rupert tunnels and soon became convinced that some shapes were Noperts. In a 2021 paper, they conjectured that the rhombicosidodecahedron is not Rupert. Their work, which preceded Murphy’s and Grimmer’s recent explorations, was, “I think, the first to conjecture that there might be solids that don’t have this property,” Steininger said. If you want to prove that a shape is a Nopert, you must rule out Rupert tunnels for every possible orientation of the two shapes. Each orientation can be written down as ...