Why π is Irrational: Mathematical Construction and Physical Reality

This paper demonstrates the irrationality of π from dual perspectives of geometric construction and physical reality. The central thesis posits that the irrationality of π is not a mathematical coincidence but a necessary condition for the logically consistent existence of the “ideal circle” as an ideal geometric object. Through retrospective analysis of the Gauss polygon approximation method, combined with the helical propagation model of photons derived from Maxwell’s electromagnetic theory, this paper reveals a profound isomorphism between mathematical irrational numbers and physical reality – the propagation of light essentially constitutes a “non-closing” circular motion, mathematically corresponding to the infinite non-repeating nature of π.