An algorithmic approach to Rupert's problem

TL;DR

This paper introduces an algorithmic method to determine whether convex polyhedra have Rupert's property, confirming it for many known solids and discovering new cases, including a counterexample.

Contribution

It proves Rupert's problem is decidable for polyhedra with algebraic coordinates and develops a probabilistic algorithm to efficiently verify the property.

Findings

Confirmed Rupert's property for all Platonic and most Archimedean solids

Discovered a new Archimedean polyhedron with Rupert's property

Provided statistical evidence suggesting the Rhombicosidodecahedron is not Rupert

Abstract

A polyhedron $\textbf{P} \subset \mathbb{R}^3$ has Rupert's property if a hole can be cut into it, such that a copy of $\textbf{P}$ can pass through this hole. There are several works investigating this property for some specific polyhedra: for example, it is known that all 5 Platonic and 9 out of the 13 Archimedean solids admit Rupert's property. A commonly believed conjecture states that every convex polyhedron is Rupert. We prove that Rupert's problem is algorithmically decidable for polyhedra with algebraic coordinates. We also design a probabilistic algorithm which can efficiently prove that a given polyhedron is Rupert. Using this algorithm we not only confirm this property for the known Platonic and Archimedean solids, but also prove it for one of the remaining Archimedean polyhedra and many others. Moreover, we significantly improve on almost all known Nieuwland numbers and finally conjecture, based on statistical evidence, that the Rhombicosidodecahedron is in fact not Rupert.