Billiards and the Five Distance Theorem II

TL;DR

This paper investigates the geometric and combinatorial properties of billiard trajectories in a rectangle, focusing on elementary segments and their weights, and provides formulas for counting segments based on certain weight relations.

Contribution

It extends previous work by deriving explicit formulas for the number of elementary segments with specific weights under certain length conditions.

Findings

Elementary segments have at most five distinct weight values.

Equal weights correspond to equal segment lengths.

Provided formulas allow calculation of segment counts given specific weight relations.

Abstract

We consider a billiard table rectangle. If a billiard ball is sent out from position F(1) at the angle of $\pi/4$, then the ball will rebound against the sides of the rectangle consecutively in points $F(2),F(3),...$. Let $n\geq5$ and $\Phi= \{F(j): 1\leq j\leq n \}$ be the set of different points. An open connected subset of the perimeter of the billiard rectangle with different endpoints from the set $\Phi$ is called \textit{segment}. \textit{Length} of a segment is a distance along the perimeter between its endpoints. A segment with endpoints $F(k)$, F(l), $1\le k,l\le n$, is called \textit{even} (or \textit{odd}), and has \textit{weight} $|k-l|$ (or $k+l$) if $k$, $l$ are of the same (or different) parity. A segment is called \textit{elementary} if there are no points of the set $\Phi$ between its endpoints. Suppose $\emptyset \neq V\subseteq\{F(1),F(n)\}$. A segment $I$ is \textit{associated} with $V$ if $I$ is an elementary segment incident with an element of $V$ or $I\cap\Phi$ is nonempty set contained in $V$. Let $\omega_1<\omega_2$ be odd weights and $\omega_0$ be an even weight of segments associated with $\{F(1)\}$, and let $\omega_3<\omega_4$ be other odd weights of segments associated with $\{F(1),F(n)\}$. Suppose that $a_i$ is the length of the segment with the weight $\omega_i$, $i = 0,..., 4$. In an earlier paper the author have proved that the weights of elementary segments have at most five different values $\omega_0,..., \omega_4$. Moreover, elementary segments with equal weights have equal lengths. Let $A_i$ be the set of all elementary segments with weight $\omega_i$. In this paper we prove that, if we know weights $\omega_0$, $\omega_1$, $\omega_4$, and $\epsilon, \delta \in \{-1, 1\}$ such that $a_2-\epsilon a_1 = a_3-\delta a_4 =a_0$, then we can easily calculate $|A_0|, ..., |A_4|$.