## 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:

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

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 {$n$}.

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

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.

Let {$m,n\in\mathbb N$} be the dimensions of a rectangle. 1. If {$m$} has more factors of 2 than {$n$}, then the path will end at corner A.
2. If {$n$} has more factors of {$2$} than {$m$}, then the path will end at corner C.
3. 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: 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.

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.