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 algebraic choice. The derivation reveals that every persistent wave is a helix, the sine wave is its one-dimensional projection, and the "imaginary" dimension is the perpendicular component of a real rotation. The π/2 result is the N=2 case of a general principle: N balanced oscillations must be equally spaced at 360°/N around the phase circle, proved via the roots-of-unity identity. The N=3 case (120° spacing) underlies the three-mode structure of Dimensional Oscillation Theory. Euler's formula e^(iθ) = cosθ + i·sinθ is interpreted throughout as a geometric description of the helix, not an abstract identity.