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)?

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, is this possible for all realizations?