Synthesis of Prime Waves on the Complex Plane and Their Symmetry: An Intuitive Model of Real-Imaginary Transitions and Cosmological Implications

This paper presents an intuitive visualization model concerning the synthesis of waves on the complex plane and the transitions between real and imaginary numbers. Using a simulation that synthesizes waves with primes p as frequencies and an amplitude of 1/(exponent 1/2), we visually demonstrate the convergence of waves on a specific symmetry plane and the formation of standing waves (cancellation of imaginary components) through the synthesis of forward and backward rotations. Furthermore, we discuss how the geometric properties of these waves extend to a three-dimensional geometry of energy—incorporating potential—and to cosmological models such as black holes acting as local origins, and the cyclical expansion and contraction of the universe.