Golden ratio and phyllotaxis, a clear mathematical link

Area of Science:

• Number Theory
• Mathematical Biology
• Plant Physiology

Background:

• Markoff's theory provides a framework for understanding rational approximations of real numbers.
• The prevalence of the golden ratio in nature suggests underlying mathematical principles governing biological growth and form.

Purpose of the Study:

• To establish a mathematical link between the difficulty of approximating real numbers and a plant's idealized growth capacity.
• To provide a theoretical explanation for the natural occurrence of the golden ratio using principles from number theory and mathematical biology.

Main Methods:

• Utilizing Markoff's theory to analyze rational approximations of real numbers.
• Defining a modular invariant function to represent plant growth capacity, dependent on the approximated number.
• Connecting the mathematical properties of number approximations to biological growth parameters.

Main Results:

• A direct correlation was established between the complexity of approximating a number and its associated idealized plant growth capacity.
• The modular invariant function demonstrated that numbers with simpler rational approximations correspond to higher growth capacities.
• The golden ratio emerged as a number with a particularly high growth capacity within this mathematical framework.

Conclusions:

• The study provides a rigorous mathematical explanation for the golden ratio's ubiquity in nature, rooted in number approximation theory.
• The concept of 'growth capacity' as a modular invariant offers a novel way to bridge abstract mathematics and biological phenomena.
• This research opens avenues for exploring other mathematical constants and their potential biological relevance through similar theoretical approaches.