Mathematical Physics

    

The Basel Problem Through Kinematic Shadows: A Kinematic-Geometric Reconstruction of ζ(2) = π^2/6

Authors: Rayan Bhuttoo

We derive Euler’s celebrated result through a novel kinematic-geometric framework. By modeling the orthogonal projection of uniform circular motion (e.g., a rotating blade under collimated light), we identify the universal ratio ∥shadow∥ circumference = 1 π as a fundamental scaling law between rotational and linear kinematics. Interpreting the real number line as a harmonic projection of a rotational system, we demonstrate that the summation reconstructs the curvature lost under projection. This approach naturally extends to higher zeta values ζ(2k), admits quantum-mechanical analogues via projection operators P, and adapts to relativistic regimes where Lorentz contraction modifies shadow geometry. Our work establishes π as a dynamic compression ratio be tween rotational and linear kinematics, offering a physical lens for classical number theory.

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[v1] 2025-09-02 20:43:24

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