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For compact four dimensional riemannian manifolds, the unique features of geometry center around the fact that the Hodge *-operator maps 2-forms to 2-forms and satisfies $*^2=1$.  For coordinates adapted to a hypersurface, the physical universe, if we write the electromagnetic field intensities as a 2-form we find that $*E = -B$ and $*B = +E$.  This relation between electric and magnetic fields and the Hodge star operator is absolutely crucial to understand that the geometry of the universe plays a crucial role on possible symmetries of macroscopic forces.  Recall that the objective evidence for four macroscopic spatial dimensions comes from crystals with rotational symmetries of orders 5, 8, 10 and 12 that have been dubbed ‘quasicrystals’.