Harmonics on the factored three-sphere and the Hopf map
  Laplacian eigenmodes on homogeneous Clifford--Klein factors of the
three--sphere are obtained as pullbacks of harmonics on the orbifolded
two--sphere using the Hopf map. A method of obtaining these polyhedral, or
crystal, harmonics using binary invariants is presented which has computational
advantages over those based on projection techniques, or those using invariants
constructed in terms of Cartesian coordinates. In addition, modes transforming
according to the irreps of the deck group are found in easy fashion using the
covariants already conveniently calculated by Desmier and Sharp and by Bellon.
Agreement is found with existing results.
