I propose an alternative version to the illumination problem (where the mirrored walls surrounding a room prevent light from reaching some region).
Here, our building area is an infinite straight band (of fixed width) on a plane that separates the two sides (two half-planes). We want to place mirrored pieces inside this building area such that:
- all incoming light from one side is reflected to that side,
- the two sides are not separated by the pieces (it should let air through).
One trivial example based on Penrose Unilluminable Room is my Mushroom-shaped design.
My question is: if the band accommodating such a wall has a unit width, then what is the widest possible narrowest gap the air has to squeeze itself through to get from one side to the other?
Fun fact: This problem was motivated by a programmer friend wanting to darken his room with a window blind while not preventing air ventilation. Of course, you can easily achieve this in practice with some black paint, but that might overheat as it absorbs most incoming light. So, the only mathematically pleasing solution is to use 100% mirrors and reflect all incoming light to the outside to let the computer scientist work in pitch darkness.
PS: Please suggest a better title/tags/definitions, if you have any ideas, I found it difficult to formulate "narrowest gap", for example.
