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In two previous papers, the electron was described in terms of a periodic structural model, namely a four-dimensional 'helix' or 'stationary wave' of spin ? symmetry. That specific model generates most first-order properties of the electron as observed, and is stable in a Casimir sense where inward vacuum pressure balances outward inertial motion. It predicts a large electrical self-repulsion equal to 1/137 of mc² across the helical diameter 2r, or a small electrical self-repulsion equal to 1/(137 ? 2p) of mc² along the curved helical path 4pr.  Here it will be shown how those two finite electrical selfrepulsions, when used together, can explain the magnetic moments of an electron or muon to second or third order in powers of 1/(137 ? p). The small self-repulsion of 1/(137 ? 2p) represents a stable part of the electron mass, and accounts for a first-order Lamb shift in atoms. By contrast, the large self-repulsion of 1/137 contributes only temporarily to electron mass, and accounts for the probability of any electron to emit or absorb light. A periodic structural model may also explain the quantized nature of magnetism in atoms, on the hypothesis that a bound electron can only join to itself using an integral number of spin  ? double-turns. The electron paths can then be considered as resonant, non-radiating rings whose net angular momenta explain the magnetic energies s, p, d, f of atomic fine-structure spectra.