## Rectangles

On the rectangle, we are interested in the path of a ball bouncing off the sides.

As on a box, we confine our attention to paths that hit the sides at an angle of {$45^\circ$}.

The behavior of billiard paths on the rectangle is well-understood. First, looking at paths starting at a corner:

**Billiard Paths Must End:**

*If the dimensions of a rectangle are positive integers, the path from any corner must end at one of the other corners.*

**Billiard Paths and the LCM:**

*Let {$m,n\in\mathbb N$} be the dimensions of a rectangle. Then the length of the path from any corner is equal to {${[m,n]}$}, the least common multiple of {$m$} and {$n$}.*

**Infinite Billiard Paths:**

*Let {$x,y$} be the dimensions of a rectangle. If {$\frac xy$} is irrational then the path from any corner is infinite.*

**Billiard Path Segments:**

*Let {$m,n\in\mathbb N$} be the dimensions of a rectangle. Then the length of the path from any corner consists of {$\frac{[m,n]}m+\frac{[m,n]}n-1$} diagonal segments.*

**Billiard Path Destinations:**

*Let {$m,n\in\mathbb N$} be the dimensions of a rectangle.*

*If {$m$} has more factors of 2 than {$n$}, then the path will end at corner A.*
*If {$n$} has more factors of {$2$} than {$m$}, then the path will end at corner C.*
*If {$n$} and {$m$} have the same number of factors of 2, then the path will end at corner B. *

Next, looking for loops, paths that don't reach corners:

**Billiard Loops and the GCD:**

*Let {$m,n\in\mathbb N$} be the dimensions of a rectangle. Then the number of loops on the rectangle is equal to {$(m,n)-1$}, one less than the greatest common divisor of {$m$} and {$n$}.*

** Corollary:**

*Let {$m,n\in\mathbb N$} be the dimensions of a rectangle. Then there are no loops on the rectangle if and only if {$m$} and {$n$} are relatively prime.*

**Summary Chart of Correspondences between Rectangles and Boxes**

Billiards on more general spaces have been studied extensively. See "Billiard dynamics: An updated survey with the emphasis on open problems" by John Smillie for the current state of research.