A Geometric Derivation of the Imaginary Unit from Three Axioms of Wave Existence

The imaginary unit i is conventionally introduced as an algebraic postulate (i² = −1). This paper derives it from geometry. Starting from three axioms about wave existence — a fundamental entity is a pure oscillation, it must persist without vanishing, and it has a definite frequency — we prove that persistence is impossible in one dimension (any real sinusoid passes through zero) and that the minimum two-component extension is unique: equal amplitudes with a phase offset of exactly π/2. The operator connecting the two components is a quarter-turn rotation; two quarter-turns make a half-turn (reversal), so it necessarily squares to −1. The property i² = −1 is a geometric consequence of wave persistence, not an…