Spiral Exponential Function — Chapter 7: Spatial Spiral Equation

This chapter extends the previously time-dependent spiral equation to spatial variables, aiming to establish a unified structural framework for divergence and rotation in three-dimensional space. A visual method is introduced by mapping the real and imaginary axes of the complex plane onto a two-dimensional real plane (Re → x-axis, Im → y-axis), thereby generalizing the direction of divergence. Geometric models such as cylinders, spheres, and infinite planes are employed to define the spiral plane, providing intuitive insight into the relationship between divergence and rotation. These extensions enable the representation of motion modes in three dimensions—something that was difficult to achieve using traditional complex numbers. In particular, the use of Clifford algebra allows for a unified expression of divergence (∇⋅F) and rotation (∇∧F), resulting in the formulation of the spatial spiral equation: This equation proposes a novel dynamical perspective: the spatial variation of the vector field f is determined by the angle-dependent structure of divergence and rotation on a rotational surface.