We present a geometric derivation of the fine-structure constant α ≈ 1/137.036 from first principles using the D3-symmetric Apollonian circle packing. Starting from a regular tetrahedron inscribed in a sphere (a natural geometric model for the electron’s spherical symmetry), we apply stereographic projection to obtain a planar configuration that, after inversion, yields the D3 Apollonian gasket. The curvatures in this packing belong to the quadratic field Q(√3) and follow a recurrence governed by the Pell equation. Within this packing, we identify a circle of curvature 92 + 26√3 whose radius (in units where the outer circle has radius 1) equals α to within 0.0002% of the experimentally measured value. This deriva-tion requires no free parameters and emerges naturally from the geometry of the tetrahedron, the most symmetric arrangement of four spheres. The result suggests that the fine-structure constant is not arbitrary but is encoded in the recursiveself-similarity of the electron’s projected spherical geometry.