  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.

Metric Geometry;Geometric Topology