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(Follow-up to this question)

Can a dodecahedron be deformed into a great stellated dodecahedron while maintaining the number of dimensions each element occupies?

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  • $\begingroup$ I did not understand your previous question, and I don't understand the new one either. $\endgroup$ Aug 31, 2021 at 17:34
  • $\begingroup$ I just want to mention that you basically ask whether the realization space of a certain matroid is connected (or at least whether two specific points belong to the same connected component). I am not an expert for this, but there might be some general theory or some algorithms that you can apply to this problem. $\endgroup$
    – M. Winter
    Sep 1, 2021 at 3:31
  • $\begingroup$ On the other question's comment thread Sam Nead asked whether you can invert a cube following the same rules. I think the answer is no: take a face F and a vertex V not on that face, then V has to pass the plane spanned by F at some point. When it does, then the two vertices sharing a face with V and two vertices of F must also lie on the same plane. And by the same argument so must the last cube vertex, so the cube is no longer 3-dimensional. $\endgroup$ Sep 3, 2021 at 1:54
  • $\begingroup$ I think the same argument should rule out a solution in the case of the dodecahedron too. That is, I don't think you can deform the dodcehadron into the GSD without a vertex passing through the plane of a neighboring face. $\endgroup$ Sep 3, 2021 at 16:46

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