  We classify the finite groups of orthogonal transformations in 4-space, and
we study these groups from the viewpoint of their geometric action, using polar
orbit polytopes. For one type of groups (the toroidal groups), we develop a new
classification based on their action on an invariant torus, while we rely on
classic results for the remaining groups. As a tool, we develop a convenient
parameterization of the oriented great circles on the 3-sphere, which leads to
(oriented) Hopf fibrations in a natural way.
