The Wager
In the 1600s, Prince Rupert of the Rhine made a bet: a cube could pass through a hole cut in an identical cube. It sounds impossible—how can you thread a shape through itself?—but Rupert was right. If you tilt a cube corner-on, its silhouette shrinks just enough that a slightly smaller cube can slip through.
Mathematicians later proved the result generalizes. A regular tetrahedron can pass through itself. So can an octahedron, a dodecahedron, a cylinder, a cone. Nearly every convex three-dimensional shape you can imagine has this property. We call it Rupert's property: a shape is Rupert if a copy of itself can pass through a straight prismatic tunnel cut in another copy.
The Cube Trick
The key insight is about shadows. Viewed from certain angles, a cube's silhouette (orthographic projection) is a regular hexagon. That hexagon is larger than the square face of the cube. So you can cut a square tunnel through the cube, and a second cube—viewed from its hexagonal angle—slips right through.
The transparent cube above has a square tunnel cut through it. The inner wireframe cube is rotated to its diagonal orientation—the angle where its shadow is a hexagon. It fits through. This is Rupert's property in action.
The Conjecture
By the early 2000s, Rupert's property had been verified for every Platonic solid, every Archimedean solid, and wide families of prisms and antiprisms. Every convex body anyone tested turned out to be Rupert.
In 2017, researchers formally conjectured that all convex polyhedra are Rupert. It seemed like a safe bet. The geometric argument feels almost obvious: for any shape, surely there exists some rotation where the shadow shrinks enough. There are infinitely many orientations. You only need one to work.
The Counterexample
In 2025, Jakob Steininger and Sergey Yurkevich proved the conjecture wrong.
They constructed a specific convex polyhedron—not exotic in appearance, almost mundane—for which no rotation allows passage. No matter how you orient it, its shadow never fits inside any other shadow of itself. The name noperthedron, coined by Tom Murphy VII, means simply: not Rupert.
The shape has 152 faces: 150 triangles and two regular 15-gons forming the top and bottom caps. These pentadecagonal caps are rotated relative to each other, and connected by belts of triangular facets that give the shape its distinctive profile—somewhere between a crystal vase and a stubby gemstone.
Shadow Projection
Rupert's property is really about shadows. A shape is Rupert if you can find two orientations where one shadow fits strictly inside the other. Then you cut a tunnel shaped like the inner shadow through the shape in the outer-shadow orientation, and the inner one passes through.
Watch the noperthedron's shadow change as it rotates. The proof shows that for this particular shape, no shadow ever fits inside any other shadow. The margins are incredibly tight—the shape is almost Rupert.
The Proof
How do you prove something is impossible for every possible orientation? Steininger and Yurkevich modeled the problem as follows. The passage of one copy through another depends on five parameters: three angles for the orientation of the “passing” copy, and two for the direction of the tunnel axis. This defines a 5-dimensional configuration space.
They divided this space into approximately 18 million tiny blocks and applied two theorems:
The computation is rigorous (not numerical approximation) and was verified independently. Every one of those 18 million blocks is certified: no passage exists.
Why It Matters
The noperthedron isn’t just a curiosity. It broke a conjecture about the universality of a geometric property. For four centuries, every convex shape anyone examined turned out to be Rupert. The intuition—that among infinitely many orientations, surely one works—felt almost self-evident.
It wasn’t. The shape exists right at the boundary: a slightly different construction might be Rupert. This one, proved across 18 million certified cells of configuration space, is definitively not.
Sometimes geometry says no.