An Investigation into the Interpretation of Pi (π) as the Arithmetic Mean of the Golden Ratio (ϕ) and the Feigenbaum Constant (δ)

Abstract

This study comprehensively examines the numerical proximity of the arithmetic mean of the golden ratio (ϕ) and the Feigenbaum constant (δ) to the number π, the theoretical mechanisms underlying this relationship, and its epistemological value. Although these three fundamental constants, situated at the intersection of mathematics and theoretical physics, are generally treated as elements of independent disciplines, this research unearths a unique structural bridge expressed through the formulation π = (ϕ + δ)/2. Numerical analyses prove that the relationship in question can be evaluated within the class of high-precision mathematical approximations, with a notably low relative error margin of approximately %0.064. Within the scope of the study, the fact that the arithmetic balance between the "most irrational number" ϕ, representing maximum dynamic stability, and the universal scaling factor δ, defining the critical threshold at which systems drift into chaos, corresponds to the number π the foundation of periodic cycles is discussed within the framework of the "geometric balance between chaos and order" hypothesis. Whether this numerical pattern is a coincidental numerological coincidence or a yet to be discovered deep topological necessity is analyzed in the context of dynamical systems theory and Euler’s identity. In conclusion, by presenting a simple and elegant equation that has gone unnoticed in the half century since the discovery of the Feigenbaum constant, this article raises new epistemological questions regarding the nature of fundamental constants and offers an interdisciplinary perspective on understanding the hidden geometry of nature.