d and f Orbital Shapes as Shadows of 4D Geometry: SO(4) Symmetry, the Fock S³ Mapping, and the Kustaanheimo-Stiefel Transform

A research note connecting the shapes of atomic orbitals to four-dimensional geometry. The hydrogen atom possesses a hidden SO(4) symmetry (Pauli 1926) arising from conservation of the Runge-Lenz vector. In momentum space, hydrogen wavefunctions are exactly the spherical harmonics on the 3-sphere S³ (Fock 1935 stereographic projection). The familiar d and f orbital shapes are therefore 3D equatorial shadows of 4D hyperspherical harmonics. Covers: the Runge-Lenz vector and SO(4) Lie algebra; the Fock stereographic projection theorem; the Kustaanheimo-Stiefel transform (3D hydrogen ≡ 4D harmonic oscillator, exact); the SO(4,2) conformal group encoding the full hydrogen spectrum in a single representation; n-sphere generalization table; and open research directions including an interactive visualization paper (Am. J. Phys. target), angular-momentum dependence of AMEM matrix elements, and lanthanide crystal field analysis in the S³ harmonic basis.