The Meaning of GRG: Extended Version with High-Precision Simulations

This work introduces the Generative Radial Geometry (GRG) method, a discrete and constructive approach to understanding the area of a circle without relying on the constant pi. GRG builds the semicircle from a rotating radius and vertical projections, summing local contributions to approximate the area. The method shows surprising accuracy even with low resolution (N = 48), and reaches near-perfect precision with higher values (N = 384 and beyond). GRG offers potential applications in education, logic, and computational geometry, and invites further exploration of pi-free geometric models.