Rhombicosidodecahedron non-Rupert conjecture

Prove that the Rhombicosidodecahedron (RID) does not have Rupert’s property. Establishing this would resolve a standing conjecture about the Rupert status of the RID.

Background

The authors previously conjectured that the Rhombicosidodecahedron fails to have Rupert’s property, but their current techniques do not resolve this case. They indicate that certain local configurations prevent a straightforward application of their local theorem, leaving the conjecture unproven.

Confirming the conjecture would extend the negative results beyond the Noperthedron, clarifying the boundary of Rupert’s property among well-known highly symmetric polyhedra.

References

In Conjecture 2 we conjectured that the Rhombicosidodecahedron (RID in short) does not have Rupert's property. We still believe this conjecture, however the methods of this paper do not suffice to prove it.

— A convex polyhedron without Rupert's property  (2508.18475 - Steininger et al., 25 Aug 2025) in Section 9: Discussion, Origin of the Noperthedron

We leave it as an open question to decide whether there exists a samurai function \sam for which

\sam(\pi(\PPP)) = \text{const}

for the Noperthedron or Rhombicosidodecahedron \PPP for any orthogonal projection \pi (possibly dropping the linearity condition in Definition~\ref{prop:sam}).

— The Nieuwland number of the Octahedron  (2609.10788 - Steininger et al., 9 Sep 2026) in Section 4, Remarks and discussion, fifth item