First Shape Found That Can't Pass Through Itself | Quanta Magazine

Abstraction: First proven convex polyhedron that cannot pass through a copy of itself

Key points:

• The Rupert property: a shape has it if a tunnel can be bored through it wide enough for an identical copy to pass; cubes, tetrahedra, dodecahedra, and many others have this property
• Jakob Steininger and Sergey Yurkevich proved a 90-vertex, 152-face shape they named the Noperthedron lacks the Rupert property — the first such shape proven to exist
• Proof required dividing orientation parameter space into ~18 million blocks, then showing each block is ruled out by either a global theorem (big mismatches) or a local theorem (small reorientations)
• The local theorem requires finding three boundary vertices whose connecting triangle contains the shadow's center, guaranteeing any small rotation pushes a vertex outward
• Decades-long conjecture that every convex polyhedron has the Rupert property is now disproved; margin for tight Rupert passages can be as small as 0.000002 times the shape's radius

Connections: Computational Geometry · Topology

Source: https://www.quantamagazine.org/first-shape-found-that-cant-pass-through-itself-20251024/