Tetrahedra passing through a triangular hole, and tetrahedra fixed by a planar frame

We show that a convex body can pass through a triangular hole iff it can do so by a translation along a line perpendicular to the hole. As an application, we determine the minimum size of an equilateral triangular hole through which a regular tetrahedron with unit edge can pass. The minimum edge length of the hole is (1+2)/6≈0.9856. One of the key facts for the proof is that no triangular frame can hold a convex body. On the other hand, we also show that every non-triangular frame can fix some tetrahedron.