It is a remarkable fact that the nonrelativistic Coulomb (hydrogenlike atom) have sets of degenerate eigenfunctions (
,
),
,
,
), and so on, whose members can exhibit completely different geometrical shapes. It was first shown by Vladimir Fock in 1935 that the Coulomb problem has the same energy spectrum as a free particle moving on the surface of a fourdimensional hypersphere. Thus the bound states of the hydrogen atom exhibit the same
symmetry as rotations in four dimensions. For the physical threedimensional problem, this is a "hidden symmetry", which can be attributed to the existence of an additional constant of the motion—the Runge–Lenz vector. In principle, the symmetry can be exhibited by stereographic projection of degenerate hyperspherical harmonics onto a threedimensional space.
This Demonstration considers a more accessible "toy" model for a twodimensional hydrogen atom, in which the degenerate orbitals can be obtained by stereographic projection of spherical harmonics on a threedimensional sphere onto a plane. The sphere can be rotated about its vertical axis by an angle of inclination
. This generates different linear combinations of tesseral harmonics
T_{ℓ m}(θ,ϕ). For example,
can be transformed into
when
is varied from 0 to
. Positive regions of the functions are shown in blue, and negative regions in green. The tesseral harmonic is projected onto the plane tangent to the south pole of the sphere. For clarity, the sphere and plane are separated in the graphic. The contours in the plane are also multiplied by an exponential factor
to show a greater range of magnitudes. The key aspect of the Demonstration is how the nodes of the tesseral harmonics, separating positive and negative regions of the wavefunction, project onto the plane. For
, we obtain a twodimensional
orbital, with the greatcircle node of the tesseral harmonic transformed into the radial node of the
orbital. For
, we obtain the angular node characteristic of a
orbital. Intermediate values of
give various

hybrid orbitals, which are also degenerate in the free atom.