Actually, the atomic orbitals aren't 4D, they are 6D, because they are complex wave functions (so each dimension is represented by a complex number, which is 2D). Of course, this is not Euclidean 6D, because while the complex numbers do behave somewhat like geometric 2D vectors, the correspondence is not 100%. So you do get effects that could be rationalized geometrically, but other effects can't.
The ring shapes of some of the d and f orbitals (and theoretically, the g orbitals) do indeed arise from toroidal loops in 6D (3D complex space) intersecting with 3D real space. But the underlying symmetry is not what you'd expect an Euclidean 6D geometry would have, since if it was actual Euclidean 6D (real space), and not just 3D complex space, it should be possible to get icosahedral orbitals by a suitable alignment of the 6 axes w.r.t. 3D real space, but as far as we know, these don't exist.