Rupert's Property

Summary

You can cut a hole in a cube that’s big enough to slide an identical cube through that hole! Think about that for a minute—it’s kind of weird. Amazingly, nobody could prove any convex polyhedron doesn’t have this property! It’s called ‘Rupert’s property’. Until this week. This week Steininger and Yurkevich proved there is a convex polyhedron that you can’t cut a hole in big enough to slide the entire polyhedron through the hole. It has 90 vertices, and apparently 240 edges and 152 faces. To prove that no such hole is possible, they had to do a computer search of 18 million different holes, plus use a lot of extra math to make sure they’d checked enough possibilities: • Jakob Steininger and Sergey Yurkevich, A convex polyhedron without Rupert’s property. To celebrate their discovery, they gave this polyhedron a silly name. Since this polyhedron lacks Rupert’s property, they called it a ‘noperthedron’. Why is this property called ‘Rupert’s property’? Wikipedia explains: In geometry, Prince Rupert’s cube is the largest cube that can pass through a hole cut through a unit cube without splitting it into separate pieces. Its side length is approximately 1.06, 6% larger than the side length 1 of the unit cube through which it passes. The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution. Prince Rupert’s cube is named after Prince Rupert of the Rhine, who asked whether a cube could be passed through a hole made in another cube of the same size without splitting the cube into two pieces. A positive answer was given by John Wallis. Approximately 100 years later, Pieter Nieuwland found the largest possible cube that can pass through a hole in a unit cube. Here Greg Egan shows how Rupert’s property works for the cube: Here he shows how it works for the regular octahedron: And finally, here’s a video by David Renshaw showing 26 polyhedra with Rupert’s property… and 5 polyhedra that might lack it: The triakis...