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A new perspective of arithmetic billiards

Yangcheng Li

TL;DR

The study analyzes arithmetic billiards on mirrored plane grids and proves that the number of closed light-paths visiting all grid points satisfies $C(m,n)=\gcd(m,n)-1$, using three distinct approaches: a Euclidean-algorithm-based argument, a diagonal-mapping construction with a fixed step length $\mathcal{K}=2\mathrm{lcm}(m,n)$, and a circular-sequence framework that yields generating-function expressions. It extends the problem to $p$-dimensional grids, yielding $\mathcal{K}=2\mathrm{lcm}(m_1,\dots,m_p)$ and $C(m_1,\dots,m_p)=2^{p-2}\frac{m_1\cdots m_p}{\mathrm{lcm}(m_1,\dots,m_p)}-2^{p-2}$, while revealing finite-abelian-group structures on the diagonal-move mappings and partitioning the grid into $2^{p-1}$ orbits. The circular-sequence method provides explicit sequence formulas and generating functions, connecting the three proofs into a unified combinatorial and number-theoretic picture. These results bridge geometric billiards with Diophantine constraints, offering a structured, high-dimensional generalization and a clear algebraic interpretation of path-counting in mirrored grids.

Abstract

We study the problem of arithmetic billiards from a new perspective. We first raise a similar problem about reflecting lights inside grids. For the solution to this problem, we will give three proofs. Next, we consider a similar problem in plane grids and give its solution. Moreover, we extend this two problems to $p$-dimensional space, where $p\geq2$. In this process, we introduce two mappings about finite discrete sets, and get two finite abelian groups. In addition, we give the definition of circular sequences and consider some combinatorial properties of circular sequences.

A new perspective of arithmetic billiards

TL;DR

The study analyzes arithmetic billiards on mirrored plane grids and proves that the number of closed light-paths visiting all grid points satisfies , using three distinct approaches: a Euclidean-algorithm-based argument, a diagonal-mapping construction with a fixed step length , and a circular-sequence framework that yields generating-function expressions. It extends the problem to -dimensional grids, yielding and , while revealing finite-abelian-group structures on the diagonal-move mappings and partitioning the grid into orbits. The circular-sequence method provides explicit sequence formulas and generating functions, connecting the three proofs into a unified combinatorial and number-theoretic picture. These results bridge geometric billiards with Diophantine constraints, offering a structured, high-dimensional generalization and a clear algebraic interpretation of path-counting in mirrored grids.

Abstract

We study the problem of arithmetic billiards from a new perspective. We first raise a similar problem about reflecting lights inside grids. For the solution to this problem, we will give three proofs. Next, we consider a similar problem in plane grids and give its solution. Moreover, we extend this two problems to -dimensional space, where . In this process, we introduce two mappings about finite discrete sets, and get two finite abelian groups. In addition, we give the definition of circular sequences and consider some combinatorial properties of circular sequences.
Paper Structure (5 sections, 22 theorems, 134 equations, 10 figures)

This paper contains 5 sections, 22 theorems, 134 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The first proof of Theorem 1.3
  3. The second proof of Theorem 1.3
  4. The generalization of the problem
  5. The third proof of Theorem 1.3

Key Result

Theorem 1.3

$C(m,n)=\gcd(m,n)-1$ holds for any $m,n\in \mathbb{N}^*$, where $\gcd(m,n)$ is the greatest common factor of $m$ and $n$.

Figures (10)

  • Figure 1: A $6\times4$ plane grid
  • Figure 2: At least $3$ beams of light needed for a $6\times4$ plane grid
  • Figure 3: A closed path in $n\times n$ plane grid
  • Figure 4: A closed path in $(m+n)\times n$ plane grid
  • Figure 5: The changing law of component $f_i(x_i)$ when $m_i=5$.
  • ...and 5 more figures

Theorems & Definitions (57)

  • Example 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : The first proof of Theorem 1.3
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.4
  • ...and 47 more