Prince Rupert's cube

The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution.

Many other convex polyhedra, including all five Platonic solids, have been shown to have the Rupert property: a copy of the polyhedron, of the same or larger shape, can be passed through a hole in the polyhedron.

One way to see this is to first observe that these four points form a rectangle, by the symmetries of their construction.

, by the Pythagorean theorem or (equivalently) the formula for Euclidean distance in three dimensions.

As a rectangle with four equal sides, the shape formed by these four points is a square.

Extruding the square in both directions perpendicularly to itself forms the hole through which a cube larger than the original one, up to side length

[1] The parts of the unit cube that remain, after emptying this hole, form two triangular prisms and two irregular tetrahedra, connected by thin bridges at the four vertices of the square.

Each tetrahedron has as its four vertices one vertex of the cube, two points at distance 3/4 from it on two of the adjacent edges, and one point at distance 3/16 from the cube vertex along the third adjacent edge.

According to a story recounted in 1693 by English mathematician John Wallis, Prince Rupert wagered that a hole could be cut through a cube, large enough to let another cube of the same size pass through it.

Wallis showed that in fact such a hole was possible (with some errors that were not corrected until much later), and Prince Rupert won his wager.

[3][4] Wallis assumed that the hole would be parallel to a space diagonal of the cube.

The projection of the cube onto a plane perpendicular to this diagonal is a regular hexagon, and the best hole parallel to the diagonal can be found by drawing the largest possible square that can be inscribed into this hexagon.

Calculating the size of this square shows that a cube with side length slightly larger than one, is capable of passing through the hole.

[3] Approximately 100 years later, Dutch mathematician Pieter Nieuwland found that a better solution may be achieved by using a hole with a different angle than the space diagonal.

Nieuwland died in 1794, a year after taking a position as a professor at the University of Leiden, and his solution was published posthumously in 1816 by Nieuwland's mentor, Jean Henri van Swinden.

[1][2][6][7][8][9][10][11][12] The construction of a physical model of Prince Rupert's cube is made challenging by the accuracy with which such a model needs to be measured, and the thinness of the connections between the remaining parts of the unit cube after the hole is cut through it.

For the maximally sized inner cube with length ≈1.06 relative to the length 1 outer cube, constructing a model is "mathematically possible but practically impossible".

[13] On the other hand, using the orientation of the maximal cube but making a smaller hole, big enough only for a unit cube, leaves additional thickness that allows for structural integrity.

[14] For the example using two cubes of the same size, as originally proposed by Prince Rupert, model construction is possible.

[16] Since the advent of 3D printing, construction of a Prince Rupert cube of the full 1:1 ratio has become easy.

is said to have the Rupert property if a polyhedron of the same or larger size and the same shape as

Cubes and all rectangular solids have Rupert passages in every direction that is not parallel to any of their faces.

[25] Another way to express the same problem is to ask for the largest square that lies within a unit cube.

More generally, Jerrard & Wetzel (2004) show how to find the largest rectangle of a given aspect ratio that lies within a unit cube.

For aspect ratios closer to 1 (including aspect ratio 1 for the square of Prince Rupert's cube), two of the four vertices of an optimal rectangle are equidistant from a vertex of the cube, along two of the three edges touching that vertex.

The other two rectangle vertices are the reflections of the first two across the center of the cube.

asks for the largest (three-dimensional) cube within a four-dimensional hypercube.

After Martin Gardner posed this question in Scientific American, Kay R. Pechenick DeVicci and several other readers showed that the answer for the (3,4) case is the square root of the smaller of two real roots of the polynomial