Maybe I’m overly annoyable right now but I was unable to find, in the first 20 or so paragraphs of fluff, a statement of what the theorem actually is, and barely a hint of even what field it’s in. Is there a better write up anywhere that actually talks about the math rather than trying to spin it into a soap opera?
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
I wonder if the bit about it only working for rational angles would be better phrased as it not being proven for these angles yet? It seems strange that it would not work for one of these if you can get arbitrarily close to it on either side and have that work.
Then again there all a lot of strange things in mathematics and my only knowledge of this problem is from this article.
That's fair to say. Rational angles happen to make things easy and result in a problem which has connections with lots of well-developed areas of mathematics.
But you also realize the dark point is arbitrarily small? It's an exact point. You can get arbitrarily close to an exact point without ever reaching it.
I also found this strange, but I think the example is much harder to understand than the article claims.
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.
I'd love to know what makes this theorem so useful, but I doubt I will understand much considering this is in an area of math I know almost nothing about.
I imagine it must be very hard to explain the theorem while making it accessible and interesting to the general public.
In the 'Numberphile' video mentioned above (about 3 minutes in), it says the initial problem about illumination was asked in the 1950s. This leads me to suspect that it has something to do with building weapons. (Brit physicist Roger Penrose found a solution where some spots were dark.)
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.
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.
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.
This over-simplification of the theorem made me think of the theory of Light Field History. In the 15th Century, Leornado Da Vinci noted "The air is full of an infinite number of radiant pyramids". Later Michael Faraday gave his “Thoughts on Ray Vibrations”, His ideas
were intended to do away with the ether in favor of lines connecting the particles; light being the vibrations of these lines or rays.Today we know it "light field' as consisting of the total of all light rays in 3D space, flowing through every point and in every direction.
So how can I design a building without doors with unilluminated spaces? What would it be like in such a space? How can we use this to design spaces where darkness is easily preserved in set spaces without adding materials? This sounds like a lovely concept for designing a temple of some kind. Or whatever building might have rooms set aside for practicing things in darkness, like playing with one's vision or learning to do chores in the dark.
I highly doubt it. Elliptic curve multiplication does involve "mirroring" the point around the curve, but it has nothing to do with real mirrors or specular reflections.
> Now place a candle in the middle of the room, one that shines light in every direction. As the light bounces around the different corners, will it always illuminate the whole room? Or will it miss some spots? A side effect of proving the magic wand theorem, Eskin said, is that it conclusively answers this old question.
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?
To help people understand why this theorem is surprising, first rephrase it from “the candle lights the entire room (bar a finite number of points)” to “the candle can be seen from any position in the room (bar a finite number of positions)”
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.
That would make a great 2D game... You’re given a floor plan at each level (with increased complexity/geometry) and you need to place the candle (light source) in a point where it will illuminate the whole floor.
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.
The only constraint on the polygon is that the angles are rational multiples. There all kinds of crazy non-convex shapes that satisfy this constraint, which is why the result is noteworthy.
I think this is also interesting when it comes to physics. A dark "point" physically means that the electromagnetic fields permeate the entire room except over a finite set of Dirac delta functions. If you were to try and put something physical in the dark point, you'd quickly turn this into a quantum mechanics problem. I don't think the dark points matter in real life because confining yourself to a single point would require infinite momentum.
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.
>Imagine a room made out of perfect mirrors, Eskin said. It doesn't have to be a rectangle; any weird polygon will do. (Just make sure the angles of the different walls can be expressed as ratios of whole numbers. For example, 95 degrees or two-thirds of a degree would work, but pi degrees would not.)
My impression was that it did not have to be a regular polygon.
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