Of course, knowing that the theorem proves that a room of mirrors is fully lit by a candle is totally worth $3M. /s
Seriously though, the article glances over the applications of such a theorem that make it so useful.
Seriously though, the article glances over the applications of such a theorem that make it so useful.
I imagine it must be very hard to explain the theorem while making it accessible and interesting to the general public.
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.
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.
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.
I suspect you need to consider perfect mirrors that don't melt, a perfect vacuum, etc..
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 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.
"Aziz! Light!~"