Cubes and Boxes have Rupert's passages in every direction

TL;DR

This paper proves that all rectangular boxes, including cubes, have Rupert's passages in every direction, confirming a long-standing geometric property and resolving assumptions about the orientation of such passages.

Contribution

It establishes that all rectangular boxes possess Rupert's passages in every direction, extending previous results and confirming assumptions about their orientation.

Findings

Rectangular boxes have Rupert's passages in every direction.

The solution to Nieuwland's problem is perpendicular to the largest square in the box.

The assumption about the tunnel's orientation for cubes is valid for all rectangular boxes.

Abstract

It is a $300$ year old counterintuitive observation of Prince Rupert of Rhine that in cube a straight tunnel can be cut, through which a second congruent cube can be passed. Hundred years later P. Nieuwland generalized Rupert's problem and asked for the largest aspect ratio so that a larger homothetic copy of the same body can be passed. We show that cubes and in fact all rectangular boxes have Rupert's passages in every direction, which is not parallel to the faces. In case of the cube it was assumed without proof that the solution of the Nieuwland's problem is a tunnel perpendicular to the largest square contained by the cube. We prove that this unwarranted assumption is correct not only for the cube, but also for all other rectangular boxes.