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Euler’s unifying algebra, geometry, and analysis in one picture. e^{ix} isn’t just an exponential; it’s a particle rotating at constant angular speed, writing cos(x) + i sin(x) in real time. One equation. Three worlds unified: • Geometry (unit circle) • Trigonometry (sin/cos projections) • Analysis (complex exponentials) The green lines? : cos(x) = (e^{ix} + e^{-ix})/2 sin(x) = (e^{ix} - e^{-ix})/(2i) This single diagram is why Fourier transforms, quantum mechanics, signal processing, and electrical engineering all work.
