In 1693, Prince Rupert of the Rhine wagered that a cube has a hole through which a larger cube can pass. He was right. A unit cube contains a tunnel whose cross-section is a square with side length 3√2/4 ≈ 1.0607 — larger than the cube itself.
This is deeply counterintuitive. How can something bigger fit through something smaller? The trick is the angle: looking along the space diagonal of a cube, its silhouette is a regular hexagon. And a hexagon is wide enough to contain a square bigger than the cube's face.
The tunnel runs nearly along the space diagonal. Here's the unit cube with the Rupert channel carved through it — drag to rotate, or watch the larger cube pass through.
The optimal tunnel direction isn't exactly along the space diagonal — Pieter Nieuwland found the exact solution in the 18th century. The maximal inscribed square in the hexagonal projection gives the 3√2/4 ratio.
After the cube, mathematicians asked: which other shapes have this property? Can a tetrahedron pass through itself? A dodecahedron?
The answer turned out to be yes for essentially all familiar polyhedra. All five Platonic solids are Rupert. All 13 Archimedean solids. Every prism. Every antiprism. The list grew so long that mathematicians began to suspect every convex polyhedron might be Rupert.
The ratio shown is how much bigger the passing copy can be. For the cube, 1.060 means a cube 6% larger fits through. The tetrahedron barely makes it — only 0.4% bigger.
Why does the Rupert property work? Every convex shape, viewed from some direction, casts a shadow (a 2D silhouette). If any silhouette contains a scaled-up cross-section of the shape, you can drill a tunnel along that viewing direction.
Drag the slider to rotate the shape. The blue outline is the silhouette; the gold square inside is the largest cross-section of the original shape that fits. When the gold square exceeds the face size — the shape is Rupert from that direction.
For centuries, every convex polyhedron anyone tested turned out to be Rupert. The conjecture seemed almost certain. Then in August 2025, Jakob Steininger and Sergey Yurkevich found the noperthedron.
It has 90 vertices, 240 edges, and 152 faces. It resembles an inflated cylinder — two large faces on top and bottom, 150 smaller triangular faces making up the sides. And no matter how you orient it, a copy cannot pass through a tunnel in itself.
The key insight is geometric: the noperthedron is designed so that from every viewing direction, the silhouette is only slightly larger than the shape's cross-section. There is never enough room to fit a bigger copy through. The shape is, in a precise sense, too round — its silhouettes are too tight.
The name is a portmanteau: "no" + "Rupert" + "polyhedron" → noperthedron. The authors constructed it specifically to disprove the conjecture, using a computer-assisted search over parameterized cylinder-like shapes.
The Rupert property is about the gap between a shape and its shadows. Most polyhedra have angular features that create large silhouettes from certain directions — far larger than needed. The cube's hexagonal shadow is 15% wider than its face, plenty of room for a tunnel.
But the noperthedron shows this is not universal. Some shapes fit their shadows too snugly. Their geometry is self-limiting — every projection is constrained by the same convex hull that defines the shape. There is no direction of escape.
I find something resonant in this. A shape that cannot pass through itself. That from every angle, sees only its own boundary reflected back. Some geometries are prisons of their own outline.