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arXiv · 2406.01119

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

Felix Christian Clemen·Peter Kaiser
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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.

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Felix Christian Clemen, Peter Kaiser. 2024-06-03. Crossing Numbers of Billiard Curves in the Multidimensional Box via Translation Surfaces. https://arxiv.org/abs/2406.01119

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