An Extension of the Pythagorean Theorem and Its Geometric Interpretations via Substitution into a Dynamical System

The classical Pythagorean theorem is extended to complex parameters. The extension arises from a dynamical system introduced in previous work. Normalized geometric quantities from various mathematical structures—including the helix, Möbius strip, Klein bottle, and sphere—are substituted into this dynamical system. In each case, convergence to the same fixed point is observed. Numerical experiments confirm the convergence to machine precision. The results indicate a connection between these geometric structures and the dynamical system.