# Golden Ratio and Phyllotaxis, what is the mathematical link?

## TL;DR

This paper establishes a mathematical connection between the golden ratio, plant growth patterns, and rational approximations of real numbers using Markoff's Theory, providing a new explanation for the golden ratio's prevalence in nature.

## Contribution

It introduces a novel link between Markoff's Theory and biological growth, offering a mathematical explanation for the golden ratio's occurrence in natural phyllotaxis.

## Key findings

- Golden ratio linked to growth capacity via modular invariant functions
- Mathematical explanation for golden ratio in natural patterns
- Connection between rational approximations and biological growth

## Abstract

Exploiting Markoff's Theory for rational approximations of real numbers, we explicitly link how hard it is to approximate a given number to an idealized notion of growth capacity for plants which we express as a modular invariant function depending on this number. Assuming that our growth capacity is biologically relevant, this allows us to explain in a satisfying mathematical way why the golden ratio occurs in nature.