The Rupert Property

The Story

1693 — Prince Rupert of the Rhine wagered that a cube could pass through a hole in another cube of the same size. John Wallis proved him right: a cube has the Rupert property.

For 300+ years, every convex polyhedron tested turned out to be Rupert. Mathematicians began to suspect all convex polyhedra might share this property.

2025 — Steininger & Yurkevich found the first counterexample: the noperthedron, a polyhedron with 90 vertices and 152 faces. No matter how you orient it, its shadow never fits inside any other shadow of itself. The proof checked 18 million parameter blocks.

* The noperthedron option shows a simplified representation. The actual noperthedron has 90 vertices and 152 faces, carefully constructed so that all projection shadows have nearly identical size.