A cube can pass through a hole bored in an identical cube. In 2025 two mathematicians built a 90-vertex solid that cannot
Bore a straight tunnel through a cube and an identical cube can slide through it. Every convex solid anyone tested could do the same trick, until 2025.
Orbit the camera to look down the tunnel, then grow the orange twin until the tunnel breaks the blue cube apart.
The story goes that Prince Rupert of the Rhine bet, in the seventeenth century, that a hole could be cut through one cube large enough for another of the same size to pass through. John Wallis showed how in 1693. The trick is that a cube looks bigger from some angles than others: seen corner-on, its outline is a regular hexagon, and a unit square fits inside that hexagon with room to spare.
A century later Pieter Nieuwland found the best possible tunnel. The largest cube that can pass through a unit cube has edge 3√2/4 ≈ 1.0607, about 6% bigger than the cube it passes through. The tunnel in the demo was found by the same search that runs in the next demo, and comes within 0.05% of Nieuwland’s value.
A solid with this property is now called Rupert. Mathematicians spent the twentieth century showing that more and more solids are Rupert. The tetrahedron and octahedron followed the cube, the dodecahedron and icosahedron were settled in 2017, and most of the Archimedean solids after that. In 2017 Jerrard, Wetzel and Yuan conjectured, a little hesitantly, that every convex polyhedron is Rupert.
Pick a solid and let the search hunt for two orientations in which one shadow fits inside the other. For the Noperthedron it never gets past 1.
A random hill climb, restarted every few hundred steps. It finds the cube’s passage near Nieuwland’s 3√2/4 ≈ 1.0607 within seconds; the dodecahedron and icosahedron only clear 1 by about 1%.
Looking straight down a tunnel turns the problem flat. Fix the tunnel’s direction. The big solid casts a shadow along it, and the twin passing through casts a shadow too. The twin fits through the tunnel exactly when its shadow fits strictly inside the big solid’s shadow. The tunnel is that shadow, extruded.
So a solid is Rupert when there are two orientations, plus a turn and shift in the plane, that put one shadow strictly inside the other. Every solid in this demo is symmetric through its centre, and for those the shift never helps: if a shifted shadow fits, so does its reflection, and by convexity so does the average of the two, which is the centred shadow. That leaves five angles: two for each orientation and one for the turn.
The number the search climbs is the Nieuwland number of an orientation pair: how much the inner shadow could be enlarged and still fit. Taking the same orientation twice always gives exactly 1, the shadow fitting its own outline. Being Rupert means beating 1 somewhere. A random search finds the cube’s 1.06 in seconds; the dodecahedron and icosahedron clear 1 by only about 1%. Searches like this cannot prove a solid is not Rupert, because it might have a passage the search missed. That needs a proof.
Three points, spun about the axis in 24° steps and reflected through the centre, make the 90 vertices. Step through the rotations and watch the solid close up.
Colours mark the three generating points: C₁, C₂ and C₃. C₁ lies exactly on the unit sphere; the other two sit just inside it, at radius between 0.98 and 0.99.
In August 2025 Jakob Steininger and Sergey Yurkevich posted “A convex polyhedron without Rupert’s property” (arXiv:2508.18475). Their solid, which they nicknamed the Noperthedron, is the convex hull of three points under a group of 30 symmetries: the fifteen rotations about one axis by multiples of 24°, each with or without reflection through the centre. It has 90 vertices, 240 edges and 152 faces, and it looks almost like a ball. That matters, because round shapes have shadows that barely change with orientation, which leaves no room to fit one inside another.
The proof splits the five-dimensional space of orientations into about 18 million small cells, after using the symmetries to cut the range of one angle by a factor of 225. Each cell is ruled out by one of two theorems. The global theorem shows that some vertex of the inner shadow lies outside the outer shadow, with an error bound covering the whole cell. The local theorem handles the hard cells where the two orientations are nearly the same and the shadows nearly coincide, where no single vertex sticks out by much.
The checks were repeated in exact rational arithmetic, so rounding errors cannot hide a passage. The same paper also exhibits a solid that is Rupert but not “locally” Rupert: its passages exist, but not near any orientation where the shadows nearly coincide. The 2017 conjecture is false, and the Noperthedron is the first convex solid proved to fail.
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