Optimal state for keeping reference frames aligned and the platonic solids

The optimal N qubit states featuring highest sensitivity to small misalignment of Cartesian reference frames are found using the quantum Cram\'er-Rao bound. It is shown that the optimal states are supported on the symmetric subspace and hence are mathematically equivalent to a single spin $J=N∕2$. Majorana representation of spin states is used to reveal a beautiful connection between the states optimal for aligning reference frames and the platonic solids.