Rupert property

A polyhedron, $P$, is said to have the  if a hole can be cut in $P$ so that an equal sized copy of $P$ can pass through the hole. For example, a 10cm &times; 5cm &times; 1cm rectangular plate can have a 5cm x 1cm slit cut parallel to its longest edges, allowing another congruent plate to pass through; this argument generalizes to prove that all cuboids with three distinct edge lengths have the Rupert property. Less obviously, the cube has the Rupert property if the hole is cut at an angle.

History
In the 17th century it was conjectured by Prince Rupert of the Rhine that a cube had the Rupert property and it was shown to be the case by the mathematician John Wallis. Prince Rupert's cube refers to the largest cube that can pass through a unit cube. Nieuwland found that the optimal Prince Rupert's cube has side length $$\frac{3\sqrt{2}}{4} \approx 1.061$$.

In 1968 Christoph Scriba proved that the tetrahedron and octahedron also have the Rupert property. In 2017, Jerrard, Wetzel and Yuan, showed that the final two Platonic solids, the dodecahedron and the icosahedron, also have the Rupert property. The conjectured that all convex polyhedra have the Rupert property. In 2021, Steininger and Yurkevich used an algorithmic approach to demonstrate that most of the Archimedean solids have the Rupert property, leaving only the rhombicosidodecahedron, the snub cube and the snub dodecahedron open. They further conjectured that the rhombicosidodecahedron does not have the Rupert property, which would make it a counter example to the previous conjecture by Jerrard, Wetzel and Yuan.

Nieuwland numbers
For a given polyhedron $P$, the Nieuwland number is the supremum of the edge ratio of the largest scaling of $P$ that can pass through a hole cut in $P$.

For example the Nieuwland number of the cube is $$3\sqrt{2}/4$$ since any cube with an edge length less than $$3\sqrt{2}/4$$, can pass through a hole cut in a cube of unit edge length. A polyhedron has the Rupert iff its Nieuwland number is &geq; 1.