Geodesic
On realizability of combinatorial types of convex polytopes over number fields
Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part V, Tome 123 (1983), pp. 203-207
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It is proved that the minimal subfield of the reals over which all real combinatorial types of convex polytopes may be realized is the field of all real algebrais numbers. 
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     author = {N. E. Mnev},
     title = {On realizability of combinatorial types of convex polytopes over number fields},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     year = {1983},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_123_a15/}
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N. E. Mnev. On realizability of combinatorial types of convex polytopes over number fields. Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part V, Tome 123 (1983), pp. 203-207. http://geodesic.mathdoc.fr/item/ZNSL_1983_123_a15/