On isometric embeddings of prisms (Q6569579)

If we stack two octahedra we get a nonconvex polyhedron with fourteen triangular faces. A paper model of the surface of this polyhedron may be refolded (creasing along different lines) into a right prism that has (up to orthogonal translation) the same end-caps. Less obviously, a polyhedron can be refolded along \textit{curved} creases to produce isometrically-equivalent surfaces made up of a finite number of smooth components: the packaging industry exploits this to make curvilinear cardboard boxes.\N\NThe authors show that a right prism can be refolded isometrically into a piecewise-smooth surface that shares, up to orthogonal translation, the original end-caps, but has arbitrarily small height -- a sort of ``accordion collapse.''