
Mathematician Wins $3M Prize for 'Magic Wand Theorem' - arunc
https://www.livescience.com/breakthrough-prize-mathematics-2019-winners.html
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mokus
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](https://arxiv.org/abs/1502.05654)

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yummypaint
Numberphile to the rescue
[https://www.youtube.com/watch?v=xhj5er1k6GQ&app=desktop](https://www.youtube.com/watch?v=xhj5er1k6GQ&app=desktop)

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Fnoord
Rooms are 3D. Why is the problem drawn out in 2D and not 3D? What difference
would it make?

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Fnoord
I would appreciate it if the downvoters would answer my question. Thank you.

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OisinMoran
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.

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notfashion
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.

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.

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curiousgal
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.

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paulpauper
why wouldn't it be lit. the photons diffuse everywhere

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SquishyPanda23
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.

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thaumasiotes
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.

~~~
SquishyPanda23
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.

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thaumasiotes
> 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.

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jonas_kgomo
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.

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crawfordcomeaux
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.

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kevinwang
Is this related to visibility problems in computational geometry (like the art
gallery problem) or am I being misled by the candle example?

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layoutIfNeeded
Yes, it’s the art gallery problem, but the walls are mirrors.

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enjoyyourlife
The paper was published in 2013

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daenz
Someone smart in math, does this have implications in elliptic curve crypto?
From what little I know about EC key generation, it seems related.

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cyphar
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.

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b_tterc_p
> 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?

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Someone
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.

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NKosmatos
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.

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crawfordcomeaux
I want to design floorplans with intentional shadow spaces.

