# Pass 3D shape through hole in scaled copy of itself

The way I understand it, the problem of Prince Rupert's cube essentially requires one to look for the projection of a cube onto $\mathbb{R}^2$ such that the largest possible square (a square being the smallest projection of a cube onto $\mathbb{R}^2$) can be inscribed inside it. That led me to the following problem: for an arbitrary connected three-dimensional geometrical object $V$, by what scale factor $s$ would $V$ have to be uniformly scaled, such that the scaled object could fit through a hole that could be cut in $V$ without making it disconnected?

Obviously, if $V$ is a cube then any $s<\frac{3\sqrt{2}}{4}$ will do and if $V$ is a sphere then any $s<1$, but I was wondering if there is some nice way to do this sort of problem for more general objects. I certainly can't see a nice way. At first I thought all that would be required would be to find which orthogonal projection of the shape onto $\mathbb{R}^2$ had maximal area, but that would not work if $V$ is not convex (e.g. a dumbbell-like shape is an extreme example, which also requires $s$ quite small) since it might not preserve connectedness, and also maximizing the area might not work, since we want to inscribe a shape within the object. Perhaps it is not even sufficient to find projections onto $\mathbb{R}^2$ since the path taken through $V$ might not be straight (e.g. passing a banana along a curved path through a hollowed-out banana). It seems like the cut itself will be hard to describe since it will have to be some surface intersecting $V$ that is given (and the scaled object must be passed from end to end 'within' this surface), and I cannot see any nice general way of working with such a surface.

My questions are: is there some clever way of working around this problem to even find a general expression for at the very least the surface required to cut $V$ such that the largest possible scaled version of $V$ can pass through it, or even an expression for $s$, and are there simpler objects other than cube and sphere (e.g. a torus, a pyramid) for which $s$ is known? More specifically, is there any way of finding for which sort of objects we can have $s>1$ (i.e. pass $V$ through a smaller version of $V$) and for which must we have $s<1$? Can $s$ be arbitrarily large for some shapes, and if not what is the largest possible value of $s$ for any shape (I assume there are shapes that require arbitrarily small $s$)? Are there shapes $V$ for which the range of allowed $s$ is discontinuous (perhaps the case for the banana example with curved path)?

• +1 for, among other things, A) making me aware that "Prince Rupert's Cube" is a thing, and B) linking to Wikipedia in case I didn't know what a banana was :) – pjs36 Jan 6 '17 at 5:32
• Thanks. I think it's pretty interesting that there are shapes with $s>1$. I also thought there might be people who'd never seen a banana :) – Anon Jan 6 '17 at 5:34
• This may help, especially its references: "Convex bodies passing through holes," Maehara, Hiroshi, and Norihide Tokushige, 2009. PDF download. – Joseph O'Rourke Jan 6 '17 at 16:37
• Also: Yap, Chee-Keng. "How to move a chair through a door." IEEE Journal on Robotics and Automation 3.3 (1987): 172-181. Cited by 37 later papers. – Joseph O'Rourke Jan 6 '17 at 16:48