We have a rectangle with it sides being a mirror. A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started. Prove that at some time the light Ray passed the center of rectangle.
This is really hard to explain without pictures, so I hope it makes any sense.
For simplicity, consider (1) the aspect ratio of the rectangle to be rational and (2) the tangent of the angle the ray forms with a wall to be rational.
(If one of these numbers were irrational, the ray would not leave the rectangle. If both numbers were irrational then either the rectangle can be scaled so they are both rational, or the ray would not leave the rectangle.)
The next thing to consider is a rectangle can be scaled without affecting the relative positions of the bounces. So we can scale the rectangle to have sides that are relative primes (a,b) and the path a 45 degrees angle. Nice and simple.
Now draw a*b reflections of the rectangle arrayed in a square ab by ab. The ray can now be drawn as a line from one corner to the opposite corner of the square.
Now the rule for which corner the ray exits.
If one of (a,b) are even, the ray exits one of the consecutive corners. It hits the center of one side when halfway there.
If both (a,b) are odd, it exits the opposite corner, passing through the very center of the center rectangle.
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Posted by Jer
on 2021-06-30 11:02:33 |
Hard to follow solution
