This is from Dylan Thurston. For some reason the mailing list decided it's spam. Rich ------ On Wed, Mar 23, 2005 at 05:57:15AM -0600, James Propp wrote:

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.) =20 Is there anything analogous for any higher-dimensional star-polytopes that would enable one to view them as Galois conjugates of convex polytopes?

Yes. For instance, in 3 dimensions, the dodecahedron is conjugate to the great stellated dodecahedron; the icosahedron is conjugate to the great icosahedron; and the small stellated dodecahedron is conjugate to the great dodecahedron. (You can get pictures of all of these at http://www.mathconsult.ch/showroom/unipoly/ ) There's also a version in 4 dimensions; I don't know about higher dimensions. I'll write up why these are conjugate when I have a minute. Peace, Dylan