Dan writes:

Hmm. Since among Euclidean spaces, only R^1 and R^2 have natural field structures, the answer to this exact question would seem to be No.

I was trying to address this issue when I wrote

(Alternatively, one can view the vertices as sitting in RxR, and hit them with simultaneous Galois actions on both coordinates.)

So for instance you could have a star-polyhedron in R^3, and replace each vertex (x,y,z) with (g(x),g(y),g(z)), where g is some Galois action on the field over Q generated by the coordinates of all the vertices. Jim