- The Noperthedron is the first convex shape proven unable to pass through a copy of itself.
- Mathematicians tested 18 million orientation blocks to rule out every possible Rupert tunnel.
- Noperts appear vanishingly rare — Murphy's algorithm found passages in hundreds of millions of random shapes.
In the 1690s, Prince Rupert of the Rhine won a bar bet by proving something counterintuitive: you can drill a hole through one cube large enough for an identical cube to slide through.
The trick involves orienting the cube along its inner diagonal, creating a hexagonal shadow wider than a square one.
For centuries, mathematicians wondered which other shapes share this "Rupert property." As Erica Klarreich's wonderful story in Quanta Magazine reports, the answer seemed to be: nearly all of them.
What is the Rupert property?
A shape has the Rupert property if you can drill a tunnel through it big enough for an identical copy of itself to pass through. It seems impossible, but most common shapes – cubes, tetrahedrons, dodecahedrons – can do this by being cleverly tilted to cast a larger shadow from certain angles.
Researchers found these 'Rupert tunnels' through dodecahedrons, icosahedrons, even soccer-ball shapes.
"I think of this problem as being quite canonical," said Tom Murphy, a Google software engineer who explored the question extensively, in a quote to Quanta. "Aliens would have come to this one."
Key figure
18 million
orientation blocks tested to prove the Noperthedron has no Rupert tunnel
When Shadows Refuse to Cooperate
The search method relies on a clever insight. Shine light through a shape from different angles and examine its shadow. If one shadow fits inside another, you can bore a tunnel that allows a copy to pass through.
With computers, mathematicians tested millions of orientations across hundreds of shapes. Most revealed Rupert passages almost immediately. A few holdouts resisted for weeks of computation.
But resistance isn't proof. There are infinitely many ways to orient a shape, and computers can only check finitely many.
The Crystal Vase That Blocks Itself
Jakob Steininger of Statistics Austria and Sergey Yurkevich of A&R Tech took a different approach.
In research posted on Arxiv, they proved mathematically that a 90-vertex shape called the Noperthedron (named after Murphy's coinage combining "Rupert" and "nope") cannot pass through itself, no matter the orientation.
The pair developed theorems to rule out entire blocks of the parameter space representing possible orientations. Their "local theorem" showed that for certain vertex arrangements, any small reorientation pushes the shadow outward, preventing passage.
Their "global theorem" quantified how far to extend each ruled-out block.
I think of this problem as being quite canonical... Aliens would have come to this one.
Tom Murphy, Google software engineer
After dividing 18 million orientation blocks and testing each, they proved every block violated either the local or global theorem. No Rupert tunnel exists.
"The natural conjecture has been proved false," said Joseph O'Rourke, emeritus professor at Smith College, to Quanta.
The Noperthedron looks like a rotund crystal vase with 150 triangles and two 15-sided polygons. One enthusiast has already 3D-printed it as a pencil holder.
A story of rare passages
This could transform the landscape. Mathematicians now know Noperts exist but remain incredibly rare–Murphy's algorithm found Rupert passages in hundreds of millions of random shapes.
The question shifts from whether impossible shapes exist to understanding what makes them so special.
Can the method identify other Noperts, or does the rhombicosidodecahedron, which resisted two weeks of computation, require different techniques entirely?
This story has many sides.
Sources
- Erica Klarreich (2025, October 24). “First Shape Found That Can’t Pass Through Itself“ Quanta Magazine
- Arxiv (2025, August 24). “A convex polyhedron without Rupert’s property“ arXiv:2508.18475v1 [math.MG]
Fact Check: Claim-by-Claim Verification Verified
15 claims verified including Prince Rupert's 1690s cube wager, the Noperthedron's 90-vertex and 152-face geometry, Steininger (Statistics Austria) and Yurkevich's (A&R Tech) proof via 18 million orientation blocks, Tom Murphy's Google affiliation, O'Rourke's Smith College commentary, and the local/global theorem methodology. One citation date corrected: arxiv paper posted August 2025, not October 2025.
Commentary
- The arxiv preprint (August 2025) awaits formal peer review, though widely discussed and accepted by the mathematics community.
- The rhombicosidodecahedron remains an unproven Nopert candidate; the method may not generalize to all holdout shapes.
- Prince Rupert's "bar bet" is a popular retelling; historical details are approximate (Wallis 1693 account).
Sources used for verification
Academic/Peer-reviewed:
Other reliable sources:
- First Shape Found That Can't Pass Through Itself - quantamagazine.org
- Prince Rupert's cube - wikipedia.org
- Slashdot coverage - slashdot.org
Fact-checked by Perplexity Sonar Pro on 2026-03-14
