An algorithmic approach to Rupert's problem - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Mathematics of Computation Year : 2023

An algorithmic approach to Rupert's problem


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.
Fichier principal
Vignette du fichier
Rupert.pdf (461.57 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03504867 , version 1 (30-01-2023)




Jakob Steininger, Sergey Yurkevich. An algorithmic approach to Rupert's problem. Mathematics of Computation, In press. ⟨hal-03504867⟩
40 View
97 Download



Gmail Mastodon Facebook X LinkedIn More