Call a set of polyhedra free if it is possible to rigidly move the polyhedra, without any polyhedron intersecting any other, so that their pairwise distances are arbitrary large, and locked otherwise. So two linked tori are locked, as is a ship in a bottle.
Can a finite set of convex polyhedra in $\mathbb{R}^3$ ever be locked?
note: We can move these polyhedra simultaneously.
On
This is an exercise 42.47 in my book. Briefly, if you are allowed to move all polyhedra simultaneously, you can always unlock them. If you are allowed to move only one at a time, you cannot (see refs in the book).
On
Summarizing some of the answers and comments given so far, with some followup on the links provided therein:
It is possible to construct a set of finitely many convex polytopes in $\mathbb R^3$ such that no single polytope can be removed from the set without at some point intersecting another polytope. This is Exercise 42.47(e) of Lectures on Discrete and Polyhedral Geometry, by Igor Pak, stated on page 376 and solved on page 409. Another such construction is illustrated on this page from "Objects that cannot be taken apart with two hands," by Jack Snoeyink ([email protected]). Another reference is: J. Snoeyink and J. Stolfi, Objects that cannot be taken apart with two hands, Discrete Comput. Geom. 12 (1994), 367–384.
It is possible to construct a set of finitely many convex polytopes such that no proper subset of them can be removed by moving the entire subset as a group ("taking apart the polytopes with two hands"). This is Exercise 42.47(h) of the above-mentioned Lectures on Discrete and Polyhedral Geometry. An example is illustrated on the above-mentioned page by Jack Snoeyink.
If we are allowed to move all polytopes independently and simultaneously, that is, for a set of $n$ polytopes we may try to take it apart with $n$ hands, any set of polytopes can be taken apart as follows. Choose a point inside each polytope. Perform a dilation of those $n$ selected points away from a common center. Translate each polytope so that the selected point within that polytope follows that dilation. (In other words, we can "explode" the set of polytopes.)
Some mechanical puzzles are manufactured that (theoretically) require $n$ hands to take apart and to reassemble.
I'm not sure whether this example is correct, but only by moving one polyhedron, we cannot "take out" anyone of them.
Regular tetrahedron has four surface. We cut away them and move outward slightly. So we have 4 piece of regular triangles in the space, they are not intersect, but they are sufficiently close. Now we can consider these triangles are polyhedron with very short thickness.
In the gap of those triangles, we put 12 pyramids pass through it. The bottom of these pyramids are bit larger than the gap, so we can't take out any pyramid. For each edge of triangles, we can set one of those pyramids lean to it. Every triangle also can't be taken out because it was stuck by 3 pyramids. So we can't take out any of them(including triangles and pyramids) only by moving itself.