Golden fraction in the theory of nucleation

TL;DR

This paper explores the prevalence of the golden fraction in nucleation size spectra, revealing its universal appearance and connection to extremum-finding methods rooted in Fibonacci numbers, supported by natural and artificial examples.

Contribution

It demonstrates that the ratio of spectrum wings in nucleation follows the golden fraction and links this to extremum search methods based on Fibonacci numbers, providing a theoretical and empirical basis.

Findings

The ratio of spectrum wings approximates the golden fraction.

Golden fraction relates to extremum-finding methods.

Proportions based on Fibonacci numbers appear in nature and human artifacts.

Abstract

The problem of the universal form of the size spectrum is analyzed. The half-widths of two wings of spectrum is introduced and it is shown that their ratio is very close to the golden fraction. In appendix it is shown that behind the golden fraction of an image one can find the information basis, i.e. the proportion of the golden fraction corresponds to some method to find extremum. The method to find extrema associated with Fibonacci numbers also leads to proportions which can be seen in nature or can be introduced artificially. The information origin of proportions is proved theoretically and confirmed by examples in nature and human life.