Take six plane mirrors and set each of them up vertically so that viewed from above they form a regular hexagon, with small gaps at the vertices. Shine a laser beam through one gap so that it emerges from the gap opposite.
By how much should the laser beam be rotated so that it once again emerges from the opposite gap, but by first reflecting just once off each of the mirrors?
| arctan | √3 11 |
Instead of reflecting the laser beam, reflect all the other mirrors in the one that the beam hits. In the reflected hexagon, the beam is a straight-line continuation of the incident beam. So the problem reduces to drawing a straight line across a honeycomb, through 7 hexagons and ending on a vertex at each end. It is then required to check the reflections of the sides to confirm that each side is crossed only once, and the beam exits through the correct vertex. The first such line exits through the wrong vertex, but the second one fits the bill.
I’ve not worked out how to produce the diagram that shows this, beyond a photo of my rather scruffy drawing.
