The Noperthedron
First convex polyhedron (90 vertices) proven to NOT have Rupert's property — cannot pass through itself.
Rupert's property: a convex body can pass a copy of itself through a straight hole cut inside itself. All Platonic solids have it. The 2017 conjecture claimed ALL convex polyhedra do. Steininger and Yurkevich constructed a 90-vertex polyhedron that provably does not, settling a 300-year-old question.
Formula
A convex body is Rupert if a congruent copy can pass through a hole cut straight through it.
Confirmed in 2024: a convex polyhedron exists that cannot pass through itself — settling a question open since Nieuwland's 1950s work.
Rupert requires a projection direction where the silhouette fits inside some cross-section — the noperthedron violates this.
Summary
Rupert's property: a convex body can pass a copy of itself through a straight hole cut inside itself. All Platonic solids have it. The 2017 conjecture claimed ALL convex polyhedra do. Steininger and Yurkevich constructed a 90-vertex polyhedron that provably does not, settling a 300-year-old question.
