See why a celebrity wandering through a mirrored hexagonal room would need 144 bodyguards to block one paparazzo from taking a snap of them, allowing for reflections. (H/T @math3ma and @emilyriehl)
Given two people (red and blue dots) between two parallel mirrors, there are an infinite number of light paths joining them, involving ever more reflections.
But *all* those paths can be blocked by obstructing just 4 points (black dots)!
This is because the virtual images of the blue dot consist of just two evenly spaced 1-dimensional lattices, and we can block *all* the midpoints between the red dot and the virtual images in each lattice with just two obstacles: one for even and one for odd displacements in the lattice.
@gregeganSF In my room I have a L-shaped mirror (the doors of a wardrobe). I can see the reflections in front and beside me. But there is a third one which no matter where I move, is stuck in the corner and always cut in two parts between the two sides.
@gregeganSF I suppose this is due to the four of "myself" being on the edges of a square so that the diagonal is always through the corner of the L. No matter where I move, I will always look at the "furthest" me across the diagonal. So in this particular case I can identify three black dots that block me looking at myself. But if someone else enters the room I bet the problem becomes more complex