In article <512E5CBC.9738E289@btinternet.com>, Frederick Williams <firstname.lastname@example.org> wrote:
> Suppose the platonic solids aren't solid at all but are made of rigid > line segments with completely flexible hinges at the vertices. The cube > can be flattened into a... um... non cube. The tetrahedron, octahedron > and icosahedron cannot be deformed at all. But what about the > dodecahedron, can it be deformed?
Here's an intuitive line of thought, not a complete proof.
Starting from a face of the cube, the four adjacent edges are parallel. That seems to be what permits the deformation. None of the other regular polyhedra has parallel edges adjacent to a face, so I suspect none can be deformed.