RectanglesOn 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.
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. 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. |