Out of curiosity I've been thinking about the following "puzzle" for a while now and maybe someone here can help.
We take a rectangle and start off at one of the corners. In that corner, which is $90^\circ$, we start drawing a line at $45^\circ$, splitting the corner into two equal parts and staying inside the rectangle with our line. As soon as the line hits an edge of the rectangle, we take a $90^\circ$ "turn" so that we stay inside the rectangle and repeat this as often as we can.
My hypothesis is that we then eventually always end up in a (nother) corner, where our problem stops as we can't take a $90^\circ$ turn there and stay inside the rectangle.
I've tried this in my head with several sizes of rectangles and it always works out, but I can't prove that it's always true for all rectangles. (also with non-integer sized rectangles, for example)
If there's anybody out there wanting to spend some time thinking about this, I would really be interested to find out the solution. :)
If we take squares, the proof is easy. Take a square with edges size 5 and give the bottom left corner the co-ordinate
(0, 0). We start a line and end up immediately at
If we take a rectangle size 6 (x-axis) by 5 (y-axis), our line "bounces" at the following points:
(0, 0);(5, 5); (6, 4); (2, 0); (0, 2); (3, 5); (6, 2); (4, 0); (0, 4); (1, 5); (6, 0) where
(6, 0) is of course a corner point.