Pi from the Pythagorean Angle Lattice: A Direct Geometric Sieve from Integer Right Triangles

Abstract

π is recovered as the ratio of two full rotations (720°) to a universal constant that emerges from the asymptotic angles of primitive Pythagorean triples. We present a rigorous derivation of the constant π strictly from the geometry of primitive Pythagorean triples. By ordering the triples according to the governing parameter n = 2a/(c−b) and evaluating the smaller acute angles θ in degrees, we prove the existence of a rigid asymptotic limit: limₙ→∞ n · θ = 229.183118052…°. We demonstrate that the ratio of the total angular measure of two complete rotations in the plane (720°) to this lattice-derived constant yields the exact value of π. This provides a direct arithmetic-geometric sieve for π, demonstrating that the constant emerges naturally from the limiting geometry of integer right triangles.