Ken Pledger wrote: > > In article <512E5CBC.9738E289@btinternet.com>, > Frederick Williams <email@example.com> 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. > > Ken Pledger.
Hmm... Suppose, instead of regular polyhedra we consider others some of the faces of which may be quadrilaterals which have no edges parallel. Such polyhedra may be deformable. So it seems to me that some property other than having parallel edges adjacent to a face is relevant. But thank you for considering the matter.
I can imagine twisting a dodecahedron so that of two parallel faces one remains fixed while the other is turned about the axis that runs through the center of them both. If I was good with my hands I'd make a model.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting