A Geometric-Exponential Construction of π Using Triangular Expansion in a Circle

This paper proposes a novel geometric method to approximate the value of π using an exponential growth process of triangular elements inscribed in a unit circle. Unlike classical approaches that rely on polygonal perimeters, calculus, or trigonometric series, this method begins with a square inscribed in the circle and recursively fills the circle with an exponentially increasing number of triangles. The resulting structure forms an infinite geometric series whose sum converges to π, revealing a new connection between exponential growth and circular geometry. We further reparameterize this process in terms of the exponential constant <i>e</i>, suggesting that π emerges as a constrained exponential limit driven by recursive geometric expansion.