A specific spiky 3D shape called a stellated tetrahedron appears to be physically incapable of passing through a hole cut in its own twin.
Rupert’s problem describes a counterintuitive geometric quirk where many 3D objects can actually slide through a hole cut in a smaller version of themselves. Most common polyhedra like cubes and pyramids allow this because of the way they can be rotated and tilted. This specific stellated tetrahedron breaks that streak, providing numerical evidence that it lacks the necessary clearance. It represents a rare geometric wall where intuition about volume and orientation finally hits a dead end. Understanding these non-Rupert shapes helps engineers determine the absolute limits of how parts can be packed or moved in tight manufacturing environments.
A stellated tetrahedron that is probably not Rupert
arXiv · 2604.26531
A convex polyhedron is Rupert if a hole can be cut into it (making its genus $1$) such that an identical copy of the polyhedron can pass through the hole. Resolving a conjecture of Jerrard-Wetzel-Yuan, Steininger and Yurkevich recently constructed a convex polyhedron which is not Rupert. We propose a search for the simplest possible non-Rupert polyhedron and provide numerical evidence suggesting that a particular stellated tetrahedron is not Rupert. The computational techniques utilize linear pr