Tetrahedra Passing through a Triangular Hole

We prove an embedding theorem that says a convex body can pass through a triangular hole∆ if and only if the convex body can be congruently embedded in a right triangular prism with base ∆. Combining this with a known result on congruent embeddings of a regular tetrahedron in a triangular prism, we show that a regular tetrahedron with unit edge can pass through an equilateral triangular hole in a plane if and only if the edge length of the hole is greater than or equal to (1 + √ 2)/ √ 6 ≈ 0.9856. For the proof of the embedding theorem we use the fact that no triangular frame can hold a convex body. On the other hand, we also show that every non-triangular frame can hold some tetrahedron, and every n-gon frame (n ≥ 4) can fix some tetrahedron.