• View
• Print
• Backlinks

Relatively prime boxes

Returning to the motivating example of billiards on rectangles, recall that

1. a pair of numbers is relatively prime iff the Euclidean algorithm ends with 0 and 1.
2. a pair of numbers is relatively prime iff the rectangle with those dimensions has no loops.

We could rewrite the first of these to

1. a pair of numbers is relatively prime iff the Euclidean algorithm ends with numbers below 2.

With that, the following definition seems natural:

Definition: For {$T\in \mathcal{B}^+$}, we say {$T$} is "relatively prime" if {$x_{_T},z_{_T}<2$}.

Definition: A "loop" on a box with dimensions {$T\in \mathcal{B}^+$} is a closed geodesic which meets every edge at an angle of 45 degrees and at a point an integral distance from the ends of the edge.

And to justify our analogy, we prove:

Relatively Prime = Loopless For {$T\in \mathcal{B}^+$}, {$T$} is relatively prime if and only there are no loops on the box with dimensions {$T$}.