Atomic Nuclei with Tetrahedral and Octahedral Symmetries

We present possible manifestations of octahedral and tetrahedral symmetries in nuclei. These symmetries are associated with the OhD and TdD double point groups. Both of them have very characteristic finger-prints in terms of the nucleonic level properties - unique in the Fermionic universe. The tetrahedral symmetry leads to the four-fold degeneracies in the nucleonic spectra; it does not preserve the parity. The octahedral symmetry leads to the four-fold degeneracies in the nucleonic spectra as well but it does preserve the parity. Microscopic predictions have been obtained using mean-field theory based on the relativistic equations and confirmed by using 'traditional' Schroedinger equation formalism. Calculations are performed in multidimensional deformation spaces using newly designed algorithms. We discuss some experimental fingerprints of the hypothetical new symmetries and possibilities of their verification through experiments.