The Platonic solids that have 6, 8, 12, and 20 sides can pass through themselves (source: David Renshaw).
For more than 300 years, geometers have studied the so-called Rupert property: whether a given convex polyhedron can have a straight tunnel bored through it such that a separate copy of the same polyhedron can pass entirely through the tunnel without splitting the host. Many classic solids, such as cubes, tetrahedra, octahedra, icosahedra, and dodecahedra, have been shown to have this property.
However, this article from Quanta Magazine reports that researchers Jakob Steininger and Sergey Yurkevich have constructed the first proven counter-example: a convex polyhedron dubbed the Noperthedron (90 vertices, 152 faces), which does not admit such a passage. Their combined use of algorithmic search, topology, and extensive computation allowed them to show that no orientation or tunnel placement permits a second copy to pass through.
Key aspects of their proof include generating an enormous parameter space of possible orientations and excluding entire “blocks” of this space using a global theorem they developed, effectively certifying that no valid tunnel exists for their shape. The construction challenges a widely held conjecture that every convex polyhedron has the Rupert property. Prior to this result, only candidate shapes resisted the property, but none had been definitively proven.
From an engineering and computational geometry perspective, this work is noteworthy because it combines discrete-geometry insight with large-scale computer search and classification. It signals that even long-standing assumptions about “tunnels through shapes” may require revision. For designers and researchers in modeling or structural computation, this result suggests that certain geometric constraints are fundamentally harder than previously assumed and that constructing or confirming the absence of a given property can demand clever algorithms and rigorous proof.