Mathematician Wins $3M Prize for 'Magic Wand Theorem'
livescience.com
livescience.com
Edit: found an explanatory paper on the theorem that appears more technical but still accessible with the right background: https://arxiv.org/abs/1502.05654
Then again there all a lot of strange things in mathematics and my only knowledge of this problem is from this article.
This paper https://www.math.brown.edu/~res/Papers/intel.pdf is mentioned in one of the Numberphile videos on the topic and it basically says this in the introduction.
By my current interpretation:
If the light shines in all directions, then a regular polygon, or in fact any convex polygon, doesn't need the mirrors to illuminate the room - the candle will light the room without needing a bounce. So, if you're thinking of a convex polygon, it's not quite relevant to this example.
The only difficult case would be if there was a concave polygon and the light had to bounce into the "caves" (or perhaps a series of rooms).
It's possible to imagine that there are specific set-ups where, if the mirrors are set up just right, the light will be bounced back out of a cave rather than filling it up.
If such a situation exists, then that bounce back would happen only with a very precise set-up: being arbitrarily close to the set-up wouldn't result in the bounce back.
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!~"
This is a point light in the middle of a regular polygon right? Why is this noteworthy? Is it that the light settles on all points evenly? Is it in spite of some sort of phase cancellation thing?
My impression was that it did not have to be a regular polygon.
This theorem assumes perfect mirrors and Newtonian light. In real life, I predict that the entire room would be illuminated. There's a path of least action for the light to take, but there is a probability of taking other paths. Scattering is not precise. I would think that would add a fudge factor to the paths.
Next, don’t think of “room”; that puts your mind too much towards simple, almost convex structures. Instead, think of the a floor of a building where all doors are removed.
For example, take the ground floor plan of the Pentagon, with its myriad of rooms and corridors, with all doors removed, and replace all walls by perfect mirrors. Is there a spot to place a candle so that it or it’s reflection, reflection of a reflection, etc. can be seen from all locations in the pentagon, bar a finite number? The theorem says there is.
Now, feel free to make it harder: add back the doors, but don’t completely close them, keeping a rational angle with the walls the door opening is in. Feel free to make the angles as small as you like.
Next, place room dividers wherever you want, as long as they are perfect mirrors, form rational angles with the walls, and don’t completely close of some room or corridor in the Pentagon.
Do you think you’ll be able to completely shield of at least one room, wherever that candle is placed? If so, you’re mistaken.
I find that so interesting, as if these positions are an intrinsic mathematical property of the room. I wonder if there's a classification of rooms this way. What do rooms that have the same number of dark points have in common?
This theorem also has implications in the limits of using sound and light for surveillance. The government could ban homes with irrational angles in order to guarantee there are no dark spots for any kind of radiation surveillance.