Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation

1. it stays a convex polytope,
2. it stays a combinatorial dodecahedron (i.e. its edge-graph does not change), and
3. all edge lengths stay the same.

Can I do this? If No, can I do it for some other realizations of the dodecahedron that is not necessarily regular? If Yes, for which other realizations of the dodecahedron is this true as well (maybe all)?