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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.

Does it make sense to ask similar questions about regular tilings of the hyperbolic plane (and higher-dimensional hyperbolic space)? This will only make sense if the vertices of such a tiling are algebraic. And that depends on what model of hyperbolic space you use. In 2 dimensions, you can use the complex structure on the inside of the unit disk, but for higher-dimensions it's probably most natural to use the parametrization of H^n by lines through R^{n+1} that intersect a hyperboloid. So then we just have to clarify what we mean by a Galois action on RP^n. There's probably a slick way to say things that I'm not thinking of right now, but if you stick to the part of RP^n where all coordinates are non-zero, then it's clear what to do (if not clear how best to describe it): e.g., in RP^2, the image of (x:y:z) should be (x':y':z') where the (well-defined) real numbers x'/y', y'/z', and z'/x' are the Galois-images of x/y, y/z, and z/x. Jim Propp