A stellated tetrahedron that is probably not Rupert

TL;DR

This paper investigates whether a specific stellated tetrahedron is Rupert, using computational methods to analyze its geometric properties and providing evidence that it likely is not Rupert.

Contribution

The authors propose a computational approach to test the Rupert property on a stellated tetrahedron and provide evidence suggesting it is not Rupert, advancing understanding of polyhedron topology.

Findings

Over 88% of a certain encoding of SO(3)×SO(3) does not allow a Rupert passage.

The stellated tetrahedron's simplicity makes computational analysis feasible.

Numerical evidence indicates the specific stellated tetrahedron is probably not Rupert.

Abstract

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.