3
$\begingroup$

Suppose I have a rectangular table, dimensions $x$ by $y$, and a billiard ball is positioned in the very center. For descriptive convenience, let us impose a coordinate system on this table with an origin (0,0) in the center of the table where I strike the ball.

Now say I strike the ball at an angle of $\theta$ with respect to the horizontal. If the ball moves forever after being struck, for what values of $\theta$ will the ball form a closed loop and eventually return to initial conditions and retrace its own path all over again, and for what value of $\theta$ will the ball not form a closed loop and never re-trace its own path?

$\endgroup$
2
  • 3
    $\begingroup$ Assume the radius of the ball is $0$, the trajectory of the ball forms a closed loop iff $\frac{x\tan\theta}{y} \in \mathbb{Q}$ $\endgroup$ Nov 16, 2013 at 8:44
  • $\begingroup$ I think Chaos Theory is relevant here, but I'm not entirely sure why. There's a Numberphile video that might talk of something similar $\dots$ $\endgroup$
    – Shaun
    Nov 16, 2013 at 8:52

1 Answer 1

2
$\begingroup$

If you unfold the billiard trajectory, the question becomes equivalent to this:

  • Suppose I have a rectangular lattice, where the unit cell has dimensions $x$ by $y$, and a billiard ball is positioned on a lattice point, which we will call the origin $(0,0)$. For what values of $\theta$ will the ball strike another lattice point $(mx,ny)$, where $m$ and $n$ are even integers?

Does that help?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.