Nothing Infinite:A geodesic passing through a point on a box with integral dimensions is either a loop or it connects in both directions to a corner. Proof: Let's say the geodesic hits one edge at a distance {$r$} from one corner.  Let's say a distance {$d$} is "{$\equiv a$}" if {$d-a$} is an integer. Then it is easy to see inductively that for every point of intersection {$p$} between this geodesic and an edge, the distance between {$p$} and a corner will either be {$\equiv r$} or {$\equiv -r$}. But there are only a finite number of such points. Therefore the geodesic is either a loop or it connects in both directions to a corner. |