Playing a game of billiard with Fibonacci
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 Lamé'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.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 DOI:
 10.48550/arXiv.1906.01911
 arXiv:
 arXiv:1906.01911
 Bibcode:
 2019arXiv190601911J
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Number Theory
 EPrint:
 12 pages, 9 figures