TITLE: Can a dodecahedron be deformed into a great stellated dodecahedron?
QUESTION [6 upvotes]: Can a convex regular dodecahedron be deformed into a great stellated dodecahedron while keeping all pentagons planar and all edges of nonzero length the whole time?

REPLY [4 votes]: Talking with Saul Schleimer, we came up with the following:
Orthogonally project the great stellated dodecahedron into the $z=0$ plane, choosing a direction that does not result in any zero length edges. Do the same for the dodecahedron.
We can realise each of these projections as the endpoint of a homotopy through affine maps $(x,y,z) \to (x,y,(1-t)z)$, so planarity is preserved.
Now that everything is in the plane, the planarity of faces is easy to maintain. Note that the condition that an edge has length zero is codimension two. Therefore we can move the vertices of the flat dodecahedron to the vertices of the flat great stellated dodecahedron in a generic way.