The Illumination Problem - Numberphile
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- เผยแพร่เมื่อ 26 ก.ย. 2024
- Featuring Professor Howard Masur from the University of Chicago. Filmed at the Mathematical Sciences Research Institute (MSRI).
Part 2 of this interview: • Problems with Periodic...
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dvd screensaver problem
AlecPlease +
+
Has no one seen the office?
no, no one has
+Arix Zajíček Hahaha.
This single point in the dark on a fully illuminated room sounds pretty amazing for a horror movie.
evil horror games be like
Except if you put someone in the room to be horrified they would act as a new scattering suface.
@@popkornking Not if they're a ghost!
Maybe like a Doctor Who episode or something
@@cheeseburgermonkey7104 Alan Wake, where the entities can only manifest in dark place.
"Yes, triangles of course are convex, thank you for that observation."
I've never heard anyone sound so sincere and so sarcastic at the same time. 😂
Welcome to maths xD
7:43 for anyone else wondering 4 years later
@@davthenub thx
@@davthenub why thank you
I jumped right to that point in the video randomly right when I read that comment. Crazy!
Buy two candles
Well all the money went to the maths degree, so the mathematicians gotta save on the candles :P
Cut the candles in half
No, buy fork handles.
@@Games_and_Music flex tape
Buy one period.
I went into Blender 3D and modeled the room with the protruding mushroom shapes. Blender uses actual ray tracing so I knew it would be fairly accurate. Sure enough nowhere that I placed a light seemed to light those sections; neat!
What happened when you put it exactly in the middle? What if you put it in one of the areas "behind" the mushrooms?
@@jakistam1000 If you put it behind a mushroom it would only light up half of a room. @TheActionlab as a video on this
@@jakistam1000if you put it behind the mushrooms, let's see the top left section, then the entire lower half will stay dark. Put it in the geometrical middle point of the shape, and the 4 behind-the-mushroom sections are dark.
The Illumination Problem
That's what we call the Minions.
Endermage77 gottem
not a problem.
Karol Nieciecki stfu
Karol Nieciecki oh nvm I didn’t see the read more my bad😂
Hahaha true
Imagine being the guy that has to lay the carpet in all these rooms?
+Azayles ha ha
You must not have watched the video :P
Give that a go! :D
Oh man. I don't even know what to say to that 😂😂
Thanks for the laugh, that's all I can say 😀
It's because of all of the angles and things. carpets usually come in square cuts. A lot of trimming and re-shaping would have to be done if you were laying the carpet down.
the room's shape is irregular. You would have to trim and do a lot more work to make the carpet fit.
0:21
This game is rigged, the holes are smaller than the ball
XD
youre not hitting the ball hard enough then
Convex, thank you for that observation.
@@oerlikon20mm29 if brute force isn't working, you're not using enough of it!
4:37 I love how the light front gets cut into smaller and smaller pieces as it travels though the room and hits corners.
Wow. That 9 minutes passed in an instant. This was so interesting.
+Nissan Karki cheers for watching
In the second figure you could have placed the light source in the middle and it would have reached every corner of the figure
Got ADD/ADHD? I have and I'm always obsessing about how long I'm able to pay attention to things and just sort of lose myself in the moment while my brain goes on autopilot.
I believe that proves that time is relative.
and 8 seconds
Oh noes. Mobs can spawn there. Light it up immediately!
Think of that in terms of why life on Earth not elsewhere near us
This would actually explain why sometimes mobs spawn in spots when we think that spot is illuminated. They're spawning on that infinitely small patch of darkness the game just doesn't have the resolution to display.
Minecraft reference
Ah yes, a fellow man of culture
Prescient, as I am watching this for the first time, and Jan 6, 2021 was only a week ago.
Wow, that animation must've taken a while
Which probably took a while to write
raytracing appeared in 80's.
I think in 2008 Intel presented a Quake engine doing real time pathtracing.
All that 3D.
xpewster I wrote in 10 minutes in JS
Carlos Jorge Pls send, I could watch these animations all day.
they just used a simple piece of code. angle of refraction=angle of incidence. as in the smallest angle from where the light hits to the surface it hit, is the same angle as frum the surface it hits to the light refracts.
Thank you for shedding light on this problem. I am much more enlightened now. The bright people at Numberphile are adding to the pool of knowledge on TH-cam and it reflects positively on our society.
Nice one.
@@fahrenheit2101 don't ya mean...
*BRIGHT* one? 😃
@@gtbgabe1478 your really reflecting the enthusiasm here
What if you
Wanted to sleep in a convex room
But mathematics said
" *You can't sleep there are monsters nearby* "
Okay, this is something that was funny
@@devincetee5335 Okay, this was something that expressed an opinion
This is actually something I've been day dreaming about for many years xD
It's fun trying to find how many bounces it takes to get from one point to another in random shapes that you find in the world.
Me too
I've done the same thing for as long as I can remember
this is what happens with the thoughts in my head...
I've also spent countless hours doing that, in both 2D and 3D environments :)
I think you might enjoy this: Interactive 2D Light Transport - benedikt-bitterli.me/tantalum/tantalum.html
Ronnie O'Sullivan would solve this, no problem.
Ronnie can put spin on photons.
paul thomas Trump is better
hheheheHEehheHEHEehehEHEH
go to truThconTesTCom< REaD THe pREseNt
Solve what??
He would get the canon shot as well 😉
I thought that said the illuminati problem
Ball Baby same
same here
Same here.
Ball Baby
same lol
they changed it
I just want to say that the academic world could certainly do with a few more great teachers like this guy. It's rare to see someone who is not only friendly, down to earth and approachable, but who seems genuinely excited by mathematics and gives the impression of simply enjoying the discussion and imparting wisdom onto students. Well done sir, you rock.
Nice but people saying genuine isnt.
I know a lot of video games (especially light engines) use this sector of mathematics for both rendering light maps and for determining vision cones for AI, but I wonder if their research into improving the fidelity of games has ever returned otherwise unknown solutions to mathematics.
I doubt it. Usually, game math is pretty straightforward. The algorithms are what's being invented. Mathematicians go to great lengths to create problems, and game developers to avoid them. Game math uses so many shortcuts and approximations.
This has happened quite a lot in the last few decades. Many big studios have research departments and there are many researchers in academia working in graphics and rendering and finding novel solutions to problems. Often they are not ground breaking discoveries, but rather tiny optimisations, unnoticed quirks/symmetries, and new applications. One big breakthrough that is is likely to come from industry is a 3d solution Navier-Stokes (used for fluid simulations) as that gets a lot of attention.
Simon Roth do you have references? i'm curious :)
for game engines, the math for this is pretty straightforward, as the guy explains.
what they do is, they usually start off with that theory and strip it down to what the engine can handle.
doesn't matter how powerful the processor is, it just won't be able to calculate every single reflection needed to achieve realistic graphics, in real time.
if you bake the lighting and shift the load over to model processing, it works, but not in complete real time.
if anything, it's maths that's always improving the games a little bit, but it will never be the other way around, because the processors just can't handle the amount of calculations.
+Dixavd, _"I know a lot of video games (especially light engines) use this sector of mathematics for both rendering light maps and for determining vision cones for AI"_
What?!? No...
anyone else see this and read "The Illuminati Problem"?
cause i didnt, i can read
yup, clickbait :(
hehehe my brain did the clickbait for them
hey im not alone
Only "those" ppl do..
ummmm.. yeah
The reason why realistic graphics are nearly impossible.
To put is a bit more general "one" reason ^^. There are so many others.
William Kappler _But_ gold is an almost perfect infrared reflector. I don't think you can get any closer than that. ( ͡° ͜ʖ ͡°)
_yess i got approval_ ahem - Why thank you for the wonderful... idea. ( ͡° ͜ʖ ͡°)
Diffraction occurs even with perfect reflectors. So, even if you have perfect mirrors that can reflect an incoming ray of light with the same angle as the incidence angle (without distorting the incident light in any way), the light still diffracts as it moves in free space. This has to do with the fact that you cannot have perfect light rays (i.e. electromagnetic plane waves) in confined space, no matter how large the space is. Therefore, in reality (or quasi-reality) with perfect mirrors, any room would still be lit at every point, even though the intensity of light would differ from point to point. Imperfect mirrors, of course, amplify this effect.
Wrong.
I keep doing these random reflection games in my head when I'm sitting in a waiting room and look at the walls, wondering how a laserbeam could bounce around them.
this video is very interesting to me. i’ve been doing this in my head for my entire life without knowing this was a real thing. i constantly do this with any shape and even faces i see. sometimes in math class i will draw a shape and bounce a line within it, circling where ever the line does not touch. great video!
0:30
Aren't reflection angles measured between the ray and the normal to the surface, not between the ray and the surface itself?
I know that in this case it doesn't matter, though I'm still curious.
I guess it was done to simplify the diagrams and explanations
Yes, as far as I know you are right. But in this case it did not matter, since it was only an example and it wasn't any complicated bend surface.
It is measured from the normal. The thing that was measured in the video is called the glancing angle.
Maybe he did it because you needn't draw the normal and hence complicate the drawing with un-necci normals..
Depends on the context. For instance, in scattering problems (specifically bragg scattering) they define their angles from parallel. It's just a nomenclature and an arbitrary choice. As long as you are consistent throughout the definitions going forward.
We usually use the angle to the normal in formulas. But the angle to the normal n and the angle to the surface s are obviously linked with 90 = n + s (in degrees) or pi/2 = n + s (in radians). From there it is easy to understand if n=n' then s=s' (for non oriented angles.)
love the background voice when its ask questions.... makes the video feels more like watching in live or in classroom
I thought the DVD Logo hitting the corner of the TV screen was cool
- OlivenickO -2 and it is. This is just cooler
Popped up in my newsfeed today that someone solved this thing (I think, at least, they called it the Magic Wand Theorem but described it in this way) and I immediately remembered this video.
Hy,, Same here. After reading the news of erik , I came here
this was really illuminating
Oh hey old me what's up?
@@justvibin1087 you're not me...
Infinitesimal point of darkness?!
I CAN'T HIDE IN THAT!
Need to skill that sneak
Imagine having a room like this and simply by moving you put someone else in the dark
Someone should design a difficult mini golf course using these polygons
Someone should biuld an impossible mini golf course like in 4:40
Hahaha that’s a great idea but I suppose it could only prevent hole in ones
@@julianrosenfeld7177 you can ask the dude to score 4 goals then :)
in a numberphile comment section are you supposed to write "first" or "1st"?
1th
Nine Four 1th
On Computerphile it really ought to be zeroth.
I second this notion.
Nine Four sqrt(0) actually
How do you prove that Tokarsky's polygon has this dark point? Is there a simple proof for it?
I think the part, that was not explained in this video is the formula that is used to test this theory. Like how he said any rational polygon only has these points, but never any areas of darkness.
Considering the video without in-depth details is already 9 minutes long, it might have just been a little too much, to explain all the really complicated stuff behind it, since the reason for the video was, to just get the gist of it.
Here's an example btw
hal.archives-ouvertes.fr/hal-00800526v2
Nice! Too bad it doesn't seem to be available for free
Ah, sorry. I'm on a university campus, so I have access to all JSTOR articles. Is there at least a preview available for the public?
*cough cough* arxiv *cough cough*
This counts just for two dimensions problems, right?
Yes, only two dimensional rooms are considered here, there are similar problems in 3d space but we won't really get much out of those until we "solve" the 2d version shown here.
One dimensional would be convex at least
Obsidian Nebula one dimentional rooms are convex by definition
no if you rotate a room that works and make it curved like it would work the same think about it any 2d slice would behave exactly like the room
I think this problem in 3 dimesions would become far more complex, and would require the use of solid angles to measure the vectors.
Numberphile is one of my favorites TH-cam channels. It is certainly the one I watch everyday. Amazing
07:24 Answer: When you ask a polygon how it's feeling and it says "circular".
Hmm, I wonder if this could have cryptographic applications with dark spots as public keys
lolorz12 That is so smart
1 dislike is from the billiard players
Ronit Mandal use curving technique, it solves the problem :v
I think the dislikes are from people who think this is clickbait.
Hi where I'm from I call it Pool :)
..then you don't know about the tables without pockets, hu? ;)
I cant stop staring at his chest hairs
Once you've seen it...
@@DylansLappalterCopium
Lol that happened to me when i read that comment
Thanks, I’ll remember this while trying to light up my rooms in Minecraf
Who would of thought that math could be fun when it's not forced down your throat and tied with exams and deadlines?
can we now talk about the overly hairy chest problem?
That can be a problem?
I looked... It cannot be unseen...
Why have you done this to me?
You notice too much
1. Why do you care about another man's chest hair?
2. He was born that way...get over it.
@@brokenwave6125 How can you be born with a jungle on your chest?
The digital effects in this video are awesome! Keep it up.
If you add a condition that light dims exponentially with travel distance, surely you'd get a smooth distribution where you could see points close by an entirely unilluminated point already be pretty dark, right?
yeah I suppose if you're traveling in a medium it would dim and also every time it bounces. you'd see more of the wave like properties of light too and if you didn't restrict it to a 2d plane I think it would be even more interesting.
Not always. In the example given in the video, light travels in a straight line to the point right next to the illuminated one. So in this example there would be a sharp step in illumination
I guess "smooth" isn't right, yeah. There will be steps. - Interestingly, that single-point-dark figure must be one of those cases where it matters whether you use an open or a closed set: If you include the boundary of the table in the table (the table is a closet set), it looks like there actually _should_ be a straight line to the dark point in question, by definition lighting it. If the border is NOT part of the area (it's open), then there is a stretch that _just barely_ occludes the point.
At least it looks like that must be the case.
I love closet sets. ( ͡° ͜ʖ ͡°)
+Kram1032 It's basically a union lines. Lines in 2d space are closed. But as we are taking the union of _infinitely many_ closed sets, we can't really predict, what the outcome is using just this basic information - it could be open, closed, both or neither and just so happens to be open within the room in the example (conversely, the set of dark spots is an intersection of infinitely many open sets that just so happens to be closed).
I just love, that just as he says "they are perminately dark" at 4:00, the light actually denies that very statement, by going to the top left square after it had been in the lower section. Sweet :)
I used to doodle this exact same thing when I was a child, only I didn't know there was maths behind it.
the best recommendation i got in youtube.
I don't even want to watch this
So.. it's basically tower defense
dedvzer no, tower defense is outside the figure. This is inside.
Depends what tower defense you are playing
And the F-117.
Why hasn't anyone made a real life version of this room!? I would pay to stand in that circle of darkness....
It doesn't work that way, is just a theoretical spot, so there is not a tiny part of the room that is in dark (7:03)
@@josiasblanco378 you can stand in the mushroom area
@@LordDoucheBags still impossible in the real world, only in theorical. since most materials do not reflex light perfectly, the mushroom area would be illuminated in a real experiment
@@JacobTheSunPreacher I think the mushroom area would be noticeably darker it just wouldn't be perfectly dark for obviously reasons. Still, I would be curious to see this effect working in real life.
I find it wild that mathematicians dedicate their lives to so many problems that don't have any clear application. Respect!
Actually, fundamental science is incredibly important. Many things that at first don't seem to have any practical application can later proof to be very important for real world problems.
I never knew this was a field in Mathematics but boy I've spent countless hours doing what this guy is doing on a paper and even just by looking at strange polygons.
You mention Roger Penrose in past-tense in this video, as near as I can tell he is still kicking
No one has seen him....must be on “that spot”
He's in that rare spot where they keep the Nobel prize medals.
But how does this make me better at pool?
You now know any ball can travel to any position on the table so if you didn't get there you just didn't reflect it enough times.
It works up to a limit, because while light can (in principle) bounce off infinitely from one mirror to another, a pool ball will slow and eventually stop.
Physics
But you should be able to hit it hard enough in most cases.
and a billiard ball can het stuck in a corner between the angle
As a sidenote: The mathematician to first come up with completely dark areas, Roger Penrose, is still alive and working, the past tense mention of him may have you believe he has passed away.
He said he "is" a mathematician and physicist.
The only past tense used is referring to his work on illumination from the past...
Imagine the applications for this. Mindblowing.
even the camera man is very sharp minded and on point with his comments.
this video was very cool.
But why dont they reflect when they hit a corner?
I also wondered about this
Because you wouldn't be able to decide on an angle of incidence. A corner is a single point, not a line like the walls, and an angle between a line and point makes no sense.
PowCrashBang Why doesn't it just bounce straight back, since the line and the point collide head on?
If it bounced straight back it would just retrace its path anyway so it wouldn't illuminate anything new and can safely be ignored.
Range Wilson but once it gets back to the origin, it would start a new path!
*_Triangles are convex, thank you for making that observation_*
Does this work in radians? Wondering because a rational number in degrees is irrational in radians.
HoxTop
He mentioned his notation:
q/p * 180°
(or whichever letters he used) to describe the angle. As I understood it, the fraction out front is what decides rational vs irrational. Radians vs degrees shouldn't matter, 180° would just be replaced by π radians.
+Mors Cornam But π is irrational, 3.141.... So my question was whether it works with p/q radians (without the π multiplication)
In the video, the constant in front is what matters when defining an angle as rational or irrational.
If the fraction is rational, the angle is rational.
If the fraction is irrational, the angle is irrational.
Units are just a matter of preference (degrees, radians, revolutions/turns) and with proper conversion factors they mean the same thing. The formula was separating units out to emphasize the fraction as the defining feature.
So, it has to be p/q * π radians? You are sure it won't work with p/q radians?
OH...That makes sense now.
Yes, it needs to be p/q * π and the fraction p/q decides rational vs irrational.
I think neat (rational) fractions of a half-turn (180°, π radians) was the convention the problem was built around.
Also, I'm not 100% sure. But I'd put my confidence level above 90%.
Really shines a light on the problem
thanks to numberphile for shedding light onto this maths problem
Ok, listen-
Now my question is: What does the person, that is in the dark, see in the mirror?
That person can't even see the mirrors.
NDos Dannyu there can't be a person in that "room" because it is 2 dimensional
if it was 3 dimensional there would be no dark spot I believe
@NDos: Welp.. Makes sense.
@Dario: I think there would be a line of dark spots.
My reasoning is for a "simple" 3D room with the 2D polynom as a crosssection everywhere (meaning a parallel floor and cieling):
If you plave the candle somewhere on the z-axis, the light would start traveling with vecotr in the z-direction too. However if we now check our room from the top, we have the 2D shape again. Light with a vecotr in z-direction is just a slower version of the light, that is traveling without a z-component.
Meaning that the solution of the 3D room is the same as the 2D one, when viewed from the top, leaving a line of dark spots.
However: This would completly change, once you introduce uneven floors or cielings...
UMos but in a 3 dimensional room there is not only walls reflecting light but also the floor and ceiling right ? That additional reflection should be enough to light every spot in the room
Mirrors work by reflecting back light that has bounced off an object back at it. Since no light is present to bouncer off the person, there's nothing to bounce off the mirror back to their eyes. Therefore, they see nothing.
I would like to have a room built in one of these shapes and stand in one of the 'dark' areas.
Nixinova Im imagining this right now :D
Wouldn't work. In the real world light is scattered and we have two eyes both of which are bigger than photons.
@@tonelemoan The Ellipse with two mushrooms would, though
Luckily we've diffraction
And light sources with non-zero width ;)
And erm, two non-zero width eyes.
Glad we have super smart people solving problems that don’t exist.
This is similar to how sound waves moves too... It has many applications!
"So triangles of course are convex. Thank you for that observation." :-D made my day
You typed FFFFFFFFFIIIIIIIIIIIIIIIIIIRRRRRRRRSSSSSSSSSSSSSSTTTTTTTTT!!!!!!!!!!!!!!!!!!!!!!!!!11111
But you were actually 68th, but you typed first.
I get that reference...
You get 5 points and a cookie :D
Rhett and Link much?
Wow, haven't watched GMM in a long time! Is it still going?
Glathir certainly, they have a new episode today
The beam of light does not "go forever unless it hits a corner". Every time light hits a mirror a small amount is absorbed, but since the light bounces around so fast the instant you turn off the light source all of the light would be absorbed.
No matter how much I love logic and physics and numbers, I still can't wrap my head around mathematical problems like this. Like, what's the point of the problem? What it's use in practical problem?
2:34
also called 'boring'
6:00 i can see he finds some humor in it. lol
2:27 "I don't have to do any off walls because I just go in a straight line. This kind of room is called C ao n v _E_ x"
Now replace the billiard ball with a DVD logo. The dark spots would be the corners.
This is not just an academic question by the way. For example, the company I work for makes cast components for aerospace. They need to be ultrasonically inspected prior to use. So the question is, are there any parts of the workpiece that cannot be inspected (I.e where the wave does not go)? If there are, the whole design must be rejected, and redesigned to ensure it can be inspected. We developed software to solve this problem. It’s quite clever, and uses the mathematics of origami.
I am actually in a class taught by Tokarsky right now
Are you though?
What are the possible applications of this by the way?
BourgealaCourge Vector graphics and lightmaps for computer-created environments, ranging from games to movies.
Didn't he say Penrose did his work in 1955 or thereabouts? One possible application, given the timing of the mid-1950s, is thermonuclear weapons. Radiation (soft X-rays, supposedly) from an exploding fission device needs to uniformly "illuminate" the fusion secondary and implode it via radiation ablation. How to get the radiation from the primary to uniformly illuminate the secondary, using some form of radiant mirrors, is very much this kind of a problem.
Geometry of a microwave oven's walls for example
pool tricks
making cool youtube videos
Interesting - touched on this stuff years ago when I tried my hand at acoustic modelling. I'll have to revisit that project
Imagine being sued by the owner of a concert hall that you designed as an architect, because you forgot to watch this video and created 'blind spots' in the audience. I think I would go hide in a Penrose mushroom if it happened to me!
I can't be the only Tim looking over these theoretical rooms, thinking, "Oooh, there's Plane Crash Corner...there's Corporate Compensation Corner..."
"huygens crying in his grave"
2:59 looks like a crying frog! "Not permitted" T.T
4:12 how to answer like a political member by answering without answering the actual question
This may help with an intuitive understanding of the problem: Rational angles usually work, because they involve a finite number of bounces; that number is an integer. With irrational angles, you might require an infinite number of bounces; that number might not be an integer... but a non-integer number of bounces would make no sense.
Who else found the animation with the reflecting light beam super satisfying?
ILLUMINATI CONFIRMED
Jet Lag NO
Too old but it worked
His chest is hairier than my future.
Your future isn't hairy tho
How do you know?
Carl from Jimmy neutron 50 years later
This channel should actually put this to an experiment and build a room with mirrors and have some sort of straight angle light source like a laser pointer to then project it across and see how much of the room is covered. That would get some views
I wonder how the shadow region evolves as we approximate Penrose's solution by a rational polygon. 🤔
yo
He's the guy that spits oceans every time he speaks
So, they are illuminating the illumination problem! Great!
It'd be really cool to see a funhouse somewhere where this was built specifically to show people that standing in a specific spot was completely dark, while stepping a little to side brought light again. Of course, it'd have to be constructed so that the point of darkness was big enough for a whole person to stand in it, which would be quite the hurdle to achieve.
It's not just quite hurdle to achieve, it's impossible. This is a mathematical problem, the point will stay infinitely small regardless of the size of the room.
I found this video very
*puts on sunglasses*
illuminating
YEEEEAAAHHH
A new branch of mathematics is needed to solve a problem like this. I get on it right away.
4:54 The idea that a single point is not illuminated is incredibly counterintuitive. This is one of the most surprising things I learned from any Numberphile video in the last few years.
I love how the one holding the camera always asks questions he knows the answers to just for the viewer. I mean I think he knows all the answers.
This video was very illuminating.
This is truly enlightening.
As an engineer, I would have to say that the room is illuminated for all practical purposes