A cube through a hole in a cube

Prince Rupert of the Rhine, a nephew of Charles I who was a cavalry commander in the English Civil War and later a scientific amateur at the Royal Society, is said to have won a wager in the seventeenth century: that a hole can be cut through a cube large enough to let another cube of the same size pass through it. It sounds impossible. A cube’s square face is as wide as the cube, and a hole the width of a face would cut the cube in two. John Wallis supplied the proof Rupert’s bet needed, and the Dutch mathematician Pieter Nieuwland, in the 1790s, found the largest cube that can pass: one $3\sqrt2/4 \approx 1.06$ times the size of the cube it passes through.

The essays before this one in the sequence that began with why the list of perfect solids stops at five were about the Platonic solids as objects of symmetry: their groups, their mirrors, and one determinant for every kaleidoscope that classifies them all. This one asks a question about them as physical shapes, about how they cast shadows, and the answer requires a search rather than a symmetry argument. All five pass through themselves. Whether every convex solid does was an open question until 2025, when one was found that does not.

The shadow along a long diagonal #

The reformulation that makes the problem tractable is about shadows. Cut a straight tunnel through a solid along some direction, and the shape of the tunnel’s cross-section, looking along that direction, must lie inside the solid’s shadow seen from that direction — the tunnel can be as large as the shadow minus a thin rim. A copy of the solid can travel through the tunnel if its own shadow, seen along the tunnel, fits strictly inside. So the question becomes: are there two directions such that the solid’s shadow along one, turned and shifted within its plane, fits strictly inside its shadow along the other?

For the cube the two directions are nearly the obvious ones. Seen face on, a cube’s shadow is a square. Seen along a long diagonal, from corner to opposite corner, its shadow is a regular hexagon, and the hexagon is wider than the square in some directions: from corner to opposite corner it spans $\sqrt{8/3} \approx 1.63$ times the cube’s edge. A square of side 1 fits inside it, tilted, with room to spare at every corner, and a square of side 1.06066 fits exactly, touching the hexagon’s sides. The best passage uses a direction slightly off the diagonal, which the search finds, and the factor it reaches agrees with Nieuwland’s $3\sqrt2/4$ to nine decimal places.

The shadow of a solid is its projection onto a plane, the operation the dot product is a shadow built from a single vector’s length times a cosine. Here it is applied to every corner of the solid at once, and the shadow is the convex hull of the projected corners. The whole problem is about two such hulls and whether one fits in the other.

The shadows a cube can cast #

The cube’s shadows are worth seeing all together, because the passage depends on choosing among them.

Face on, the shadow is a unit square. Seen through the middle of an edge it is a rectangle, $\sqrt2$ by 1. Tilted in a general direction it is a hexagon, and along a long diagonal it is the regular hexagon, whose area is $\sqrt3 = 1.732$. There is a tidy rule behind the areas: the shadow is made of the three faces turned towards the viewer, each foreshortened by the cosine of its tilt, so the area is the sum of the three direction cosines, which runs from 1 to $\sqrt3$. A larger shadow is a necessary condition for a tunnel, and the hexagonal shadows are up to 73 per cent larger than the face. But area is not enough: the square must fit in shape, and the hexagon is wider than the square only along certain lines, which is why the square has to be turned to sit across the hexagon’s widest diagonal rather than along its sides.

A dust that almost every line misses used shadows in the opposite way, averaging a set’s shadow over every direction to measure how often a random line meets it. The cube’s average shadow, by the same averaging, is $3/2$ — Cauchy’s formula makes the average shadow of any convex solid a quarter of its surface area — and Rupert’s problem asks not for the average but for one particular pair of shadows, the best-placed, to beat the shape of a face.

Fitting one polygon inside another #

Given the two directions and a turn of the inner shadow, deciding how large the inner shadow can be made and still fit inside the outer one is an exact calculation. The outer shadow is a convex polygon, so it is the set of points satisfying one linear inequality per edge. Scaling the inner polygon by a factor $\lambda$ and shifting it by $(t_x, t_y)$ keeps it inside exactly when, for each edge of the outer polygon, the inner polygon’s farthest point in that edge’s outward direction stays behind the edge — one inequality per edge, each linear in $\lambda$, $t_x$ and $t_y$.

Maximising $\lambda$ subject to a handful of linear inequalities in three unknowns is a linear programme, the problem the lines the optimum lies under set up for production and diets. Its optimum lies where three of the inequalities hold with equality, so it can be found exactly by trying every triple of edges, solving the three equations, and keeping the largest $\lambda$ that satisfies all the others. For the shadows of Platonic solids there are at most ten edges, a hundred and twenty triples, and the answer comes out to full machine precision.

What remains is choosing the directions and the turn: five numbers in all, two angles for each direction and one for the turn. The factor is a complicated, piecewise smooth function of those five numbers, with many local peaks, and there is no formula for its maximum. A search finds it: start from random directions and turns, climb by repeated small adjustments until no adjustment helps, and keep the best result from many starts. The directions found are stored with the figures, and every figure recomputes the passage from them, so each factor drawn is verified by the exact linear programme rather than quoted from the search.

All five Platonic solids pass through themselves #

The same search, run on each Platonic solid, finds a passage for every one.

The cube and the octahedron both reach $3\sqrt2/4 = 1.06066$. That they agree is no coincidence: the octahedron is the cube’s dual, its corners at the centres of the cube’s faces, as the five solids as three groups arranged them, and the two share their symmetry group, though why the dual pair should have exactly the same factor is a fact found here rather than a theorem quoted. The other three manage only about one per cent: the tetrahedron 1.0145, the icosahedron 1.0108 and the dodecahedron 1.0108. Christoph Scriba showed in 1968 that the tetrahedron and octahedron pass through themselves; the dodecahedron and icosahedron were settled only in 2017, by Richard Jerrard, John Wetzel and Liping Yuan, by searches of the same kind as this one.

The margins are small, and the figures show why the question took so long for the last two. For the dodecahedron and icosahedron the outer and inner shadows are both roughly round many-sided polygons of nearly the same size, and the passing copy only just fits; a slightly worse choice of directions leaves it a fraction of a per cent too large. Nothing about the solids’ symmetry announces that a passage exists. It has to be found.

The hole has to be cut on the slant #

For the cube it is instructive to watch the factor change as the tunnel’s direction moves. Start with the tunnel running straight through two opposite faces and tilt it, along a great circle, towards a long diagonal.

Face on, the cube’s shadow is the square itself, and a square of the same size fits only by coinciding with it: factor exactly 1, no room to spare, no passage. As the tunnel tilts, the shadow widens into a hexagon, the square can be turned to use the new width, and the factor rises, peaking at 1.0409 about 45 degrees from the face normal, before the hexagon becomes regular at the long diagonal, 54.7 degrees, and the factor falls back slightly. Even the best point on this arc is short of Nieuwland’s 1.0607. The best tunnel lies off the arc altogether, tilted sideways as well, and finding it means searching in more directions than one curve can show.

The tilt is the point of Rupert’s wager. A hole straight through the faces cannot work, because a cube’s face is exactly as wide as the cube; the gain comes entirely from the extra width of the oblique shadow, and only a slanting tunnel can use it.

Passages have to be searched for #

How hard is a passage to find? Run a simple hill-climbing search from many random starting directions and turns, and record where each one stops.

For the cube, 90 per cent of the searches end in a passage, and many reach close to the maximum; the cube’s shadows are varied enough that almost any start can climb to a fit. For the dodecahedron only half the searches find a passage at all, the rest stopping at local peaks just below 1, and the best of these forty searches reaches 1.0097, short of the 1.0108 found by longer searches. The landscape of the problem, factor as a function of five angles, is full of local maxima below 1, and a passage, when it exists, sits at the top of a narrow ridge. That is why the dodecahedron and icosahedron resisted proof until searches were run with care, and why a solid with no passage would be hard to recognise by search alone.

Why the cube has so much room #

The cube’s generous 6 per cent and the dodecahedron’s thin 1 per cent have a geometric explanation. A passage needs one shadow of the solid to be substantially narrower, in some direction, than another of its shadows. The cube’s shadows vary a great deal: face on it is a square of width 1, and along a diagonal it spans 1.63, so there is a large difference to exploit. The dodecahedron and icosahedron are much rounder — their twenty and twelve corners sit close to a sphere, and their shadows from every direction are many-sided polygons of nearly the same size and shape. A round solid casts nearly the same shadow from everywhere, and a copy of nearly the same shadow fits inside it only by a sliver.

That reasoning points the same way as the counterexample. A solid that is round enough — many faces, every shadow nearly the same disc — should have no passage at all, since nothing can fit strictly inside its own copy when all its copies look alike from every side. The Noperthedron is built to be round in exactly this sense while still having flat faces, and its proof had to show that the residual variation in its shadows is never enough. The tetrahedron is the opposite case: very far from round, yet its factor is only 1.0145, because its shadows, though varied, are triangles and quadrilaterals whose narrow and wide directions do not line up conveniently. Roundness is a guide, not a rule, and the search is what decides.

The sphere is the limiting case. Every shadow of a ball is the same disc, and a copy of the ball can never pass through a hole in another of the same size: a ball has no passage, trivially, and a solid with no passage is in that sense a polyhedron that has managed to behave like a ball.

A solid that cannot pass through itself #

In 2017 Jerrard, Wetzel and Yuan conjectured that every convex polyhedron has Rupert’s property: that for any solid with flat faces, some tunnel admits a copy of the same size. Passages were found, solid by solid, for many of the Archimedean solids, the thirteen thirteen more when one word is dropped obtained by relaxing the Platonic definition, and for many other families. The conjecture held for every solid anyone tested. It was always a conjecture about convex solids; the star polyhedra of the four that are allowed to cross themselves pass through their own faces, and a tunnel through one of them is not even a well-defined thing, so the question has only ever been asked of solids without dents.

In 2025 Jakob Steininger and Sergey Yurkevich disproved it. They constructed a convex polyhedron, which they called the Noperthedron, and proved that no passage exists: for every pair of directions and every turn, the inner shadow fails to fit strictly inside the outer one. A search can only ever find passages, never prove their absence, so their proof had to cover the whole five-dimensional space of directions and turns at once. They divided it into a very large number of small regions and showed, region by region, with bounds on how fast the shadows can change within a region, that none of them contains a passage — a computer-assisted argument in which the computer checks inequalities rather than searching for examples.

What the pictures cannot show #

Every factor drawn is a lower bound: the passage it describes exists, verified by the exact linear programme, but the search that found it may have missed a better one. For the cube and octahedron the factor matches Nieuwland’s value, which is known to be the maximum; for the other three solids the figures show the best this search found, and whether larger factors exist is a question about the global maximum of a function the search samples. The shadows are drawn in their planes, not as tunnels through solids, and the actual hole — the region removed from the solid — is not drawn; it is the outer shadow’s region pushed through the solid along the tunnel’s direction, minus enough rim to keep the solid in one piece.

The figure of local searches is a sample of one simple method from random starts, and the percentages depend on that method; a better search would find more passages and higher factors. Nothing in these pictures shows a solid without a passage, because nothing a search produces can.

Still open: which solids pass through themselves #

Which convex polyhedra have Rupert’s property? Since 2025 the answer is known not to be “all of them”, and no simple criterion is known that decides it. A solid with many nearly equal faces, like a finely faceted ball, should fail, since its shadows in every direction are nearly the same disc and nothing fits strictly inside its own copy; a solid with flat, wide faces and sharp corners, like a cube, passes easily. Between those extremes, whether a given solid passes is decided today only by searching for a passage or by a large computation proving there is none.

The Archimedean solids, and the Catalan solids that are their duals, have been examined one solid at a time by searches like this one, which can confirm a passage but never rule one out. Whether a solid as symmetric as the Platonic ones can fail to pass through itself, as the Noperthedron does, or whether high symmetry always leaves room for a slanting tunnel, is the question Rupert’s bet leaves after three and a half centuries.