This paper provides a geometric derivation of the atomic wavefunction and the shape of electron orbitals. We propose that the s, p, d, and f orbitals are not probabilistic charge clouds, but the stereographic projections of deterministic standing waves vibrating on a 5-dimensional toroidal manifold. We demonstrate via computational simulation that the standard "Quantum Numbers" (n, l, m) correspond directly to topological Winding Numbers on the torus. By modulating the major radius of the torus by harmonic integers, we reproduce the exact topology of the sphere (s-orbital), dumbbell (p-orbital), and cloverleaf (d-orbital). This work reinterprets quantum nodes as regions of "Dimensional Transmutation," identifying the physical origin of inertial mass as motion rotated into the scalar dimension.