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The cube trick that became a conjecture

A cube can perform a geometric trick that seems impossible at first: it can pass through a straight hole cut into an identical cube. The opening does not simply copy one of the cube’s square faces. By tilting the moving cube and choosing the hole’s orientation carefully, the passage can be made just large enough. The question dates to Prince Rupert of the Rhine in the 17th century, and shapes able to pass through an identical copy eventually became known as “Rupert.”

Mathematicians found the same property in many other three-dimensional shapes. The pattern was so persistent that researchers formally conjectured in 2017 that every convex polyhedron is Rupert. Convex means that the solid has no inward dents; polyhedron means that its surface consists of flat polygonal faces. Together those conditions describe a broad, orderly family of solids, making the proposed universal rule plausible.

But a universal claim can be defeated by one counterexample. In “Shape Shift,” Emma R. Hasson reports that mathematicians Sergey Yurkevich and Jakob Steininger have supplied exactly that: the first convex polyhedron proved unable to pass through a hole in an identical copy.

Building a shape that cannot pass itself

The new solid has 90 vertices, 240 edges and 152 faces. It is called the “noperthedron,” using a term coined in 2025 by independent computer science researcher Tom Murphy VII for a shape that is not Rupert. Other polyhedra had been suspected of lacking Rupert’s property, but suspicion was not proof. Yurkevich and Steininger designed their object with features that made an exhaustive argument possible.

The result also completes a line of work the pair began years earlier. They met as teenagers preparing for a math olympiad and later became collaborators. After encountering Prince Rupert’s cube as university students, they recognized that nobody knew how common the property really was. In a 2020 paper they were the first to publicly conjecture that some convex polyhedra are not Rupert. Five years later, their construction established that claim.

That chronology matters. The noperthedron is more than an unusually complicated solid: it overturns the stronger 2017 conjecture that every convex polyhedron can pass through itself. Mathematics often advances this way. A long run of positive examples suggests a sweeping pattern; a carefully engineered exception then exposes the boundary of the pattern and replaces a yes-or-no question with sharper ones about which structural features make the difference.

Turning continuous motion into a finite check

Proving that the noperthedron works is harder than drawing it. To show that a solid is Rupert, researchers need only exhibit one suitable opening and one successful orientation. To show that it is not Rupert, they must rule out every possible shift and rotation. A failed physical trial proves almost nothing because a better alignment might still exist.

Yurkevich and Steininger converted this continuum of possibilities into a five-dimensional parameter cube encoding possible holes and orientations. They then combined geometric reasoning with a custom computer program to eliminate every region of that space. The program was not merely testing a large collection of promising positions; it supported an exhaustive argument that no placement allows one copy of the polyhedron to pass through the other.

An outside mathematician who studies Rupert’s property, Pongbunthit Tonpho of Chulalongkorn University, described the approach as creative and rigorous and expressed surprise that the conjecture was disproved so quickly. Still, the Scientific American article is a compact report on a preprint. It does not present the technical proof, the verification details or a peer-review history, so readers must look to the underlying paper for those questions.

The lasting significance is conceptual. The noperthedron closes one conjecture while opening a richer classification problem: what distinguishes self-passable convex polyhedra from those that remain trapped by their own geometry? Its name is playful, but its role is serious. It shows how computation can help certify a counterexample across a space of continuous possibilities - and how a single strange shape can redraw the limits of a whole geometric family.