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Mathematician Wins $3M Prize for 'Magic Wand Theorem'
- b_tterc_p 7y ago> 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?
- RcouF1uZ4gsC 7y ago>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.
- ironSkillet 7y agoThe 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.
- readams 7y agoRational angles and only a finite number of points are not illuminated. You can still get points that are dark, but not regions.
- anonytrary 7y agoI 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.
- aidenn0 7y agoIt's trivially true for a convex polygon, but non-obvious for many concave polygons.
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- Someone 7y agoTo 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.
- popcorncolonel 7y agoGreat visualization, thanks!
- NKosmatos 7y agoThat 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.
- crawfordcomeaux 7y agoI want to design floorplans with intentional shadow spaces.
- pbhjpbhj 7y agoThere's a good game called Lazer Maze (by ThinkFun) that is related, it concerns shining a directional Lazer though.
- daenz 7y agoSomeone smart in math, does this have implications in elliptic curve crypto? From what little I know about EC key generation, it seems related.
- cyphar 7y agoI 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.
- OisinMoran 7y agoI 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.
- notfashion 7y agoThat'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. This paper https://www.math.brown.edu/~res/Papers/intel.pdf 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.
- anonytrary 7y agoBut 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.
- learnstats2 7y agoI 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.
- curiousgal 7y agoOf 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.
- andrewcchen 7y agoI'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.
- p1esk 7y agoHe said it takes him “hours” to explain the theorem to phds in other subfields of math.
- 8bitsrule 7y agoIn 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.)
- AstralStorm 7y agoAmong other things, completeness of simplex, trust-region like optimization, problem space search algorithms.
- tezka 7y agohow is it related to any of these? do you have links to articles?
- paulpauper 7y agowhy wouldn't it be lit. the photons diffuse everywhere
- enjoyyourlife 7y agoThe paper was published in 2013
- mokus 7y agoMaybe 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 https://arxiv.org/abs/1502.05654
- paulpauper 7y agoreally fascinating stuff. thanks for sharing / the article does not do justice at all to the concept., read the pdf and google it instead.
- yummypaint 7y agoNumberphile to the rescue https://www.youtube.com/watch?v=xhj5er1k6GQ&app=desktop https://www.youtube.com/watch?v=xhj5er1k6GQ&app=desktop
- samasblack 7y agoAnother great explanatory paper written for a broad audience is http://www-personal.umich.edu/~alexmw/BilliardsToModuli.pdf http://www-personal.umich.edu/~alexmw/BilliardsToModuli.pdf
- kevinwang 7y agoIs this related to visibility problems in computational geometry (like the art gallery problem) or am I being misled by the candle example?
- layoutIfNeeded 7y agoYes, it’s the art gallery problem, but the walls are mirrors.
- crawfordcomeaux 7y agoSo 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.
- jonas_kgomo 7y agoThis 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.