The Fine Structure Constant as a Geometric Angle

Bohr’s 1913 atomic model treated the electron as a dimensionless point. A decadelater, Compton scattering revealed that electrons possess spatial extent characterised by thereduced Compton wavelength ¯λc. This paper demonstrates that the fine structure constantα is the geometric consequence of placing an extended electron on a circular orbit.When an object of length ¯λc sits on a circle of radius a0, it subtends an angle α =¯λc/a0. This is not an analogy; it is what α is—an angle in radians measuring the electron’sangular extent on the Bohr orbit. The constant appears throughout atomic physics becauseevery calculation must account for this topological fact: the electron is not a point but afinite arc on its own orbit.The extended electron requires an angular correction. A point particle closes on acircle after traversing 2π radians. But an electron with angular extent α must traverse2π +α to return to its starting configuration. This correction factor of α/2π is precisely theanomalous magnetic moment first calculated by Schwinger, for which he shared the 1965Nobel Prize. Schwinger’s famous result g = 2(1 + α/2π) is not a quantum field theoreticabstraction—it is the geometric correction required when an extended object must closeconsistently on its orbit.I derive a formula expressing α entirely through two measurable lengths:α = 2√(πλ̄c / λLy)where λLy is the Lyman limit wavelength. No electromagnetic constants appear—nocharge, no permittivity, no Planck’s constant. The input wavelengths are independentlymeasurable through Compton scattering and hydrogen spectroscopy. The formula repro-duces the CODATA value to ten significant figures.The fine structure constant is not a free parameter. It is fixed by the requirement thatan electron of extent ¯λc must close consistently on an orbit of radius a0 + α/2π. Thequestion “why 1/137?” becomes: why does the electron have this extent relative to itsorbit? The mystery moves from an unexplained number to a geometric constraint with