[math-fun] Galois-actions and star-polytopes

The regular pentagon and pentragram can be viewed as Galois conjugates of one another, under the imbedding in C in which all vertices are fifth roots of unity. (Alternatively, one can view the vertices as sitting in RxR, and hit them with simultaneous Galois actions on both coordinates.) Is there anything analogous for any higher-dimensional star-polytopes that would enable one to view them as Galois conjugates of convex polytopes? Jim Propp