You may be imagining a simple room. There are lots of polygonal rooms (with non mirror walls) where a candle in the center leaves dark areas.

It's not obvious (to me anyway) that it's impossible to construct a sufficiently pathological room of mirrors such that there is at least one dark corner.

You are correct. https://en.wikipedia.org/wiki/Illumination_problem

Wow cool thanks for the link!

It seems to me that the amount of lighting in the room will depend on (1) the luminosity of the candle, and (2) the interior volume of the room. Add all the mirrors you want, but you're not going to light a large area with one candle because you don't have enough light.

I don't know this research, but I think the candle metaphor shouldn't be taken too far.

The candle is probably more like a source of particles and the claim is that if the particles reflect off the boundary and you can send particles from every direction from the center, then you come arbitrarily close to any other point in the room.

Or something along those lines. Again I stress I don't know the research and haven't seen the paper and I'm just speculating.

> then you come arbitrarily close to any other point in the room

It looks like the claim is that all but a finite number of points are actually achieved. This is a stronger result -- you can easily prove from that claim that you can get arbitrarily close to any point (because a neighborhood of any radius contains an infinite number of points, and therefore contains a point that is achieved), but you can't go the other way. It's quite possible to be arbitrarily close to every point while failing to hit an infinite number of points.

The candle is infinitely luminous, and yet, the mirrors won't melt

Are you an engineer?

I suspect you need to consider perfect mirrors that don't melt, a perfect vacuum, etc..

I don't think perfect mirrors affect the result. The problem statement reminded me that there is a strain of astronomy that wonders why the sky is dark at night, given that, as best we can determine, no matter what direction you're looking in, you're looking directly at a star distant from you.

Of course that star is generally very far away. An ordinary inverse-square law has no trouble explaining why the influence of the light it's shining on you is negligible. A stream-of-particles theory of light has more trouble explaining that.

This result shows that at any point in a mirrored room, there is always a path that allows you to "directly" view any other point in the room. The more perfect the mirrors are, the more direct your view is; absolutely perfect mirrors bring you up to, but not beyond, the limit case where you're just looking at the candle without benefit of mirrors, as is the case for stars. But (I assume) the result doesn't show that a candle is actually able to illuminate a room regardless of luminosity (and, by implication, that the night sky is bright rather than dark).

If reflection was diffusive it would be more or less trivial.

If the room is connected, the is a path from the source A to any chosen point B (by definition). Now take this path and modify it such that it is composed of straight lines, and may touch walls. This new path is a diffuse light path from A to B.

Specular reflection seems much more difficult.

The room can be any polygon, including shapes where there are walls occluding other walls from the candle's light

Prove it.