Pythagorean Triples and the Circle

The universal limit arising from all Pythagorean triples, when multiplied by the right angle, yields the circle: We prove that primitive Pythagorean triples encode the circle in their universal limit. For any sequence of primitive Pythagorean triples with governing parameter n = 2a/(c−b) → ∞, the product nθ converges to 4, where θ is the smaller acute angle measured in radians. This limiting value of 4, if read as an angle in radians, equals 720°/π = 229.183118052°. Together with the elementary observation that 4 × π/2 = 2π, this reframes the value 4 obtained from the triple limit as the natural link between the discrete geometry of integer right triangles and the continuous geometry of the circle.