Contents

# Contents

## Idea

$\phi \;\coloneqq\; \tfrac{1}{2} \left( 1 + \sqrt{5} \right)$

is called the golden ratio (sometimes the golden mean).

The golden ratio is an algebraic integer and generates the ring of algebraic integers in the quadratic subfield $\mathbb{Q}(\sqrt{5})$ of the cyclotomic field $\mathbb{Q}(\zeta)$ of fifth roots of unity. It also generates the group of units (modulo its torsion subgroup) of the ring of algebraic integers $\mathbb{Z}[\phi]$.

The golden ratio is of course an irrational number, and is distinguished among irrational numbers by having the slowest convergence rate of rational approximants obtained by truncating its continued fraction expansion

$\phi = 1 + \frac1{1 + \frac1{1 + \frac1{1 + \ldots}}}$

Its rational approximants $p/q$ are ratios of successive Fibonacci numbers.

It is widely believed that this slowest rate of convergence explains observations of phyllotaxis in botany, i.e., the arrangement of leaves and plant structures such as the scales of pinecones and pineapples, in which nascent leaves or buds tend to form at the least crowded available spots on the growing plant, whereupon in the ensuing jostling the structures become arranged as “Fermat spirals” with a Fibonacci number of structures along each spiral.