Pi (π), Metallic Means, and the Pythagorean Angle Lattice

π is derived from the geometry of Pythagorean triples and the algebraic structure of classical metallic means : We present a precise derivation of the constant π from two related sources: the Pythagorean Angle Lattice and the metallic means. For a primitive Pythagorean triple with smaller acute angle θ and governing parameter n = 2a/(c − b), the identity cot(θ/4) = Mn is established, where Mn = (n + √(n² + 4))/2 is the n-th metallic mean. From the Pythagorean Angle Lattice, we obtain the limit lim (n→∞) 720°/(nθ) = π. Using the identity cot(θ/4) = Mn, this is equivalently expressed as lim (n→∞) 180°/(n · arccot(Mn)) = π. Thus the constant π is obtained directly from the geometry of the lattice as well as from the algebraic structure of the metallic means. Numerical evidence is provided.