# Playing a game of billiard with Fibonacci

## TL;DR

This paper explores the connections between billiard dynamics in a square, Fibonacci numbers, and the Euclidean algorithm, providing geometric interpretations and new constructions of the golden ratio.

## Contribution

It introduces a novel geometric framework linking billiard maps, Fibonacci sequence, and the Euclidean algorithm, rederiving Lamé's theorem with a new perspective.

## Key findings

- Rederived Lamé's theorem using geometric methods.
- Established connections between gcd properties and Fibonacci numbers.
- Presented a new way to construct the golden ratio geometrically.

## Abstract

By making use of the greatest common divisor's ($gcd$) properties we can highlight some connections between playing billiard inside a unit square and the Fibonacci sequence as well as the Euclidean algorithm. In particular by defining two maps $\tau$ and $\sigma$ corresponding to translations and mirroring we are able to rederive Lam\'{e}'s theorem and to equip it with a geometric interpretation realizing a new way to construct the golden ratio. Further we discuss distributions of the numbers $p,q\in \mathbb{N}$ with $gcd(q,p)=1$ and show that these also relate to the Fibonacci sequence.