Arithmetic billiards

Arithmetic Billiard

Goal: Arithmetic billiards are a geometric construction which allow an interesting visualisation for the greatest common divisor and the least common multiple of two positive integers. For an introduction to the subject, see this article. We will consider the following generalization: Considering three (or even more) integers i.e. the trajectory of a ball bouncing inside a parallelepiped rather than a rectangle.

Collecting experimental data will allow to make conjectures concerning the shape of the path, which then one can try to prove. It is possible that the key ideas used for the base case can be extended to prove the conjectures.

Arithmetic Billiard

Supervisors: Antonella Perucca, Sebastiano Tronto

Difficulty level: Bachelor thesis / EML 2,3

Tools: Any computer language suffices to describe the integer coordinates (however, a geometric visualisation could be very helpful).