I was thinking when about to watch a movie the other day about the logo that one often sees, saying something such as "Sony Home Video", moving in lines around the screen and bouncing at the boundaries. It is a humorous but surprisingly entertaining thing to watch it move and see if it ever hits the corner exactly.

Naturally (at least, for a recreational mathematics enthusiast), the question arose: Given the dimensions of a rectangle and the coordinates of a point inside the rectangle, along with a direction of travel for that point, creating a path where the point, on reaching a side of the rectangle bounces of off it like a light beam, can you create a method of determining whether (given infinite time) the path eventually hits one of the corners exactly?

Equally, and perhaps more interestingly, can you display a construction of a set of initial conditions for a path (for any rectangle) that is not periodic (i.e. if it hits a specific point on the boundary of the rectangle, that point never appears in its path again)?

EDIT: In light of @dtldarek's observation, it has become clear that if $x_0, y_0$ are the initial coordinates of the point $p$ in the rectangle with gradient $g\in\mathbb{R}$ in the rectangle with sidelengths $s_0, s_1\in\mathbb{R}$, then the path eventually hits a corner if and only if $\exists n, m \in \mathbb{N}$ and $k\in\mathbb{R}$ such that $$kg+(x_0,y_0) = \frac{ns_0}{ms_1}$$

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    $\begingroup$ I wouldn't be surprised if picking any two random points on the boundary as the starting point and the first bounce-point is aperiodic with probability $1$. $\endgroup$ – dtldarek May 12 '18 at 16:17
  • $\begingroup$ @dtldarek: Neither would I, as I suspect it is to do with some simple observation about using irrational angles and coordinates.... $\endgroup$ – Isky Mathews May 12 '18 at 16:18

For a rectangle with sides $a$ and $b$ create a rectangular grid with the same ratio. Then, if you draw an infinite line, then its path is exactly the path of bouncing path only that some of the rectangle you pass are reflected.

reflection demonstration

In other words, the bouncing path hits a corner if and only if the infinite straight line hits a grid-point.

I hope this helps $\ddot\smile$

Edit: I still need to think about periodicity.

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    $\begingroup$ The infinite grid observation is nice! However, the statement "... it will never be periodic if all the ratios (rectangle and path direction) are incommensurable" is not true. Consider the rectangle with $y$-axis side of length 1 and $x$-axis side $\sqrt{2}$ and the point which starts at the bottom left corner and has a gradient in its path of $\frac{1}{\sqrt{2}}$ -- this simply bounces back and forth between the bottom-left and top-right corners and is most definitely periodic. $\endgroup$ – Isky Mathews May 12 '18 at 16:40
  • $\begingroup$ What I think you meant was that the path is not periodic if the ratio of the side lengths of the rectangle is rational but that the gradient of the initial path is irrational and vice-versa. $\endgroup$ – Isky Mathews May 12 '18 at 16:42
  • $\begingroup$ Also the starting coordinates are rational... $\endgroup$ – Isky Mathews May 12 '18 at 16:45
  • $\begingroup$ However, in the general case, it is really not clear still (i.e. with potentially all irrational stuff). I have edited the main post to mention your observation. Nice diagram BTW! $\endgroup$ – Isky Mathews May 12 '18 at 16:58
  • $\begingroup$ @IskyMathews $1/\sqrt{2} = \sqrt{2}/2$ and so the ratio of the gradient and the ratio of the rectangle are commensurable. $\endgroup$ – dtldarek May 12 '18 at 17:00

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