Unavoidable golden ratio

TL;DR

This paper demonstrates that iterative geometric constructions inevitably converge to the golden ratio, potentially explaining its widespread appearance in ancient structures, natural growth, and self-organization processes.

Contribution

It introduces a simple iterative geometric process that unavoidably leads to the golden ratio regardless of initial proportions.

Findings

Iterative geometric procedures converge to the golden ratio.

Convergence is rapid and independent of initial ratios.

The results may explain the prevalence of the golden ratio in nature and history.

Abstract

It is demonstrated that iterative repeating of some simple geometric construction leads unavoidably in the limit to the golden ratio. The procedure appears to be quickly convergent regardless of a ratio of initial elements sizes. This could explain a widespread occurrence of the golden ratio in various constructions, including puzzling proportions encountered in structures of ancient cultures (e.g., in the Great Pyramid of Giza), as well as in the natural growth and self-organization processes in organic and inorganic matter.