My question is: how to find the coordinates of the vertices of a dodecahedron?
I can find the coordinates of the vertices of a tetrahedron as the solutions of a certain polynomial system in 8 variables (see tetrahedron.mw for details).
However, that approach seems not to work for a dodecahedron. A new idea is required.
PS. Of course, I have in mind a regular dodecadron.