A stellated tetrahedron that is probably not Rupert
This work addresses the open problem of finding the simplest non-Rupert polyhedron, but the result is incremental as it only provides numerical evidence for a specific candidate.
The authors propose a search for the simplest non-Rupert polyhedron and provide numerical evidence that a particular stellated tetrahedron is not Rupert, showing that over 88% of a certain encoding of SO(3) × SO(3) does not yield a Rupert passage.
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 program solvers to compute the largest possible scalings of polygons that can be translated to fit in other polygons. The relative simplicity of the stellated tetrahedron as compared to other polyhedra allows this more rudimentary check to be computationally tractable. In particular, we show that over 88% of a particular encoding of $\text{SO}(3) \times \text{SO}(3)$ equipped with the standard measure does not yield a Rupert passage.