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Euler’s Quaternion Formula

Euler’s Quaternion Formula is the 4D generalization of Euler’s classic complex identity, showing that every rotation in 3D space is actually a smooth trajectory on the hypersphere S3. In this visualization, the quaternion q = cos ϕ + u sin ϕ traces the full geometry of orientation: the real axis bends into the imaginary 3-space, the rotation axis u shapes the direction, and the angle ϕ unfolds cleanly along a normalized 3-sphere. This is the mathematical engine behind spinors, SU(2) symmetries, and the double-cover structure that makes quantum states fundamentally geometric.** In Resonant Quantum Mechanics (RQM), this formula isn’t just a tool, it’s the core of how physical systems organize. Quaternions describe orientation on the manifold S3×Rs, and Euler’s Quaternion Formula is the exact map that turns rotation into curvature. By visualizing the formula directly, you’re seeing the underlying geometry of resonance, coherence, and quantum behavior as RQM understands it. This is the hypersphere that every qubit, spinor, and eigenstate truly lives on