Crossing Numbers of Billiard Curves in the Multidimensional Box via Translation Surfaces

TL;DR

This paper investigates the crossing numbers of billiard trajectories in multidimensional boxes with commensurable side lengths, revealing a surprising power-of-two pattern in intersection counts through algebraic and number theoretic methods.

Contribution

It introduces a novel connection between billiard curve intersections and solutions to a constraint satisfaction problem, using translation surfaces to analyze multidimensional billiard dynamics.

Findings

Number of intersections is either 0 or a power of 2.

Established a bijection between intersections and constraint satisfaction solutions.

Applied algebraic and number theoretic tools to billiard dynamics.

Abstract

The billiard table is modeled as an $n$-dimensional box $[0,a_1]\times [0,a_2]\times \ldots \times [0,a_n] \subset \mathbb{R}^n$, with each side having real-valued lengths $a_i$ that are pairwise commensurable. A ball is launched from the origin in direction $d=(1,1,\ldots,1)$. The ball is reflected if it hits the boundary of the billiard table. It comes to a halt when reaching a corner. We show that the number of intersections of the billiard curve at any given point on the table is either $0$ or a power of $2$. To prove this, we use algebraic and number theoretic tools to establish a bijection between the number of intersections of the billiard curve and the number of satisfying assignments of a specific constraint satisfaction problem.