Two Sufficient Conditions for a Polyhedron to be (Locally) Rupert

TL;DR

This paper establishes two new sufficient conditions under which a convex polyhedron can be considered Rupert, meaning it can be oriented and slightly altered to fit inside a projection of itself, supporting the conjecture that all convex polyhedra are Rupert.

Contribution

The paper introduces two novel sufficient conditions for a polyhedron to be Rupert, focusing on simple orientations and local passages, advancing the understanding of the Rupert property.

Findings

Two new sufficient conditions for polyhedra to be Rupert.

Development of a theory of local passages involving small rotations.

Simplification of the problem by focusing on local alterations and orientations.

Abstract

Given two cubes of equal size, it is possible - against all odds - to bore a hole through one which is large enough to pass the other straight through. This preposterous property of the cube was first noted by Prince Rupert of the Rhine in the 17th century. Surprisingly, the cube is not alone: many other polyhedra have this property, which we call being Rupert. A concise way to express that a polyhedron is Rupert is to find two orientations $Q$ and $Q'$ of that polyhedron so that $\pi(Q)$ fits inside $\pi(Q')$, with $\pi$ representing the orthogonal projection onto the $xy$-plane. Given this scheme, to bore the hole in $Q'$ we can remove $\pi^{-1}(\pi(Q))$. There is an open conjecture that every convex polyhedron is Rupert. Aiming at this conjecture, we give two sufficient conditions for a polyhedron to be Rupert. Both conditions require the polyhedron to have a particularly simple orientation $Q$, which we alter by a very small amount to get $Q'$ as required above. When a passage is given by a very small alteration like this, we call it a local passage. Restricting to the local case turns out to offer many valuable simplifications. In the process of proving our main theorems, we develop a theory of these local passages, involving an analysis of how small rotations act on simple polyhedra.