The topic dates back 333 years, to when Prince Rupert bet that a cube could pass through a hole cut in an identical cube; he was right, and for a while, it even seemed like every single convex polyhedron might be Rupert. But some resisted extensive random searches, and Steininger and Yurkevich conjectured that the Rhombicosidodecahedron (RID), a highly symmetric Archimedean solid, might be one of the exceptions. Last year they constructed the Noperthedron, the first polyhedron proved Nopert, built specifically for that purpose.
I have now proved their conjecture for the RID. It is the first ever non-synthetic polyhedron to provably have this property, and only the second ever overall. The proof is computer-assisted, a branch-and-bound elimination with exact arithmetic in Q(√5): a certificate of 192696 regions covers the whole symmetry-reduced configuration space and takes only a few CPU minutes to generate and check. One of the main tools is the zoom lemma, which resembles blow-ups in algebraic geometry.
It took 2 years and 1000+ hours of self-motivated research. I submitted the preprint to arXiv on 29 September (still in their queue), but it can be found on Zenodo: https://zenodo.org/records/23013945
Code and certificate: https://github.com/bence-hervay/nopert-rid
