Two years ago I asked a question about finding couples of integers (p,l) such that one may draw l lines on the euclidean plane, creating exactly p intersection points, and I got very interesting comments and references.
(Numbers of intersection points and lines)
I'd like now to discuss the problem on the torus ; so far, it seems to me that the problem has not been investigated in this framework.
Here are a few really trivial comments as an introduction. With 0 and 1 line, there is 0 point of intersection. With 2 lines, one may have any integer as the number of intersection points (take an horizontal line and a correctly chosen slanted line). With 3 lines, it is possible to draw lines such that there are exactly 1, 3, or any even number of intersection points. The most generalized result I have found is that, given n lines, for any k between 0 and n, one may draw k parallel lines and (n-k) other parallel lines, creating this way exactly k(n-k)j intersection points, where j may be any integer (consider for starting figure k horizontal lines and (n-k) vertical lines for the k(n-k) result, then replace vertical lines by slanted lines correctly chosen).
Is it possible to say more? It's up to you now, thanks by advance for any comment :)
