The Illumination Problem - Numberphile
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- čas přidán 27. 02. 2017
- 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.
"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
@@davechen4979 thx
@@davechen4979 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.
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.
Lasertag strategy guide: Where to stand If your opponent can't move.
Parker Lee lol
break their legs and run
Actually this might be useful for video game spawning
Behind a wall lol
Cripple laser tag. Lets go!
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
4:37 I love how the light front gets cut into smaller and smaller pieces as it travels though the room and hits corners.
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.
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
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!
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.
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
1:25 "that is a funny shaped room." While as billiard table it was perfectly normal? Where does this guy play billiard?
I immediately thought of Lunar Pool for NES.
Amogus
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 CZcams 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
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.
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...
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.
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 would love to see someone actually build a physical demo of this. Does it still work in 3D space, I wonder?
Light bends in real life, air is imperfect and distorts and lenses the light, walls will have imperfections, no mirror perfectly reflects light, so it couldn't be done in the real world.
@@overestimatedforesight Remove the air and the very concept of up and down.
@@overestimatedforesight I think the Penrose example would be worth a try, with a strong light source, long exposure, and very very vertical walls. Maybe the distortion and scattering aren't so bad that you completely illuminate those squares, maybe you even see the boundary, and that would be cool for teaching. But it's a question of whether this can really demonstrate the math; if the image converges on the expectation for 1 -- 100 reflections, what's to say it doesn't all get ruined at reflection 101?
Global Illumination Exists so no
@@Solesteam so build it in space!
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 😉
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!
this was really illuminating
Oh hey old me what's up?
@@justvibin1087 you're not me...
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
I thought that said the illuminati problem
Ball Baby same
same here
Same here.
Ball Baby
same lol
they changed it
love the background voice when its ask questions.... makes the video feels more like watching in live or in classroom
Numberphile is one of my favorites CZcams channels. It is certainly the one I watch everyday. Amazing
I used to stare at the old Windows 98 screen saver which was bouncing around on the edges of the screen.
I'm not the only one, right ?
+
No, you are not.
There's a video of it bouncing in the corner on youtube, search for it.
DVD players too
911gp Still to this day I haven't found better things to do.
i did too, but then i changed OS
Brady, you have a knack for asking the perfect questions during the interview. Bravo.
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
The digital effects in this video are awesome! Keep it up.
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.)
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 love this channel so much. One of the only channels I'll watch at normal speed.
I thought the DVD Logo hitting the corner of the TV screen was cool
- OlivenickO -2 and it is. This is just cooler
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*
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
I'm absolutely astonished by the fact that the unreachable point is somewhere in the middle rather than some 'hidden' corner. Amazing work!
Imagine having a room like this and simply by moving you put someone else in the dark
Infinitesimal point of darkness?!
I CAN'T HIDE IN THAT!
Need to skill that sneak
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? ;)
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!
Thanks, I’ll remember this while trying to light up my rooms in Minecraf
I cant stop staring at his chest hairs
Once you've seen it...
@@DylansLappalterCopium
Lol that happened to me when i read that comment
07:24 Answer: When you ask a polygon how it's feeling and it says "circular".
What an animation! Great job!
even the camera man is very sharp minded and on point with his comments.
this video was very cool.
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 :)
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).
This is truly enlightening.
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 :)
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.
the best recommendation i got in youtube.
I don't even want to watch this
Really shines a light on the problem
Top notch visuals in this video!
Hmm, I wonder if this could have cryptographic applications with dark spots as public keys
lolorz12 That is so smart
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?
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.
This video was very illuminating.
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.
Mind blowing stuff sir
Imagine the applications for this. Mindblowing.
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
Very nice and easy-to watch video.
Really cool animations for this!
*_Triangles are convex, thank you for making that observation_*
6:00 i can see he finds some humor in it. lol
thanks to numberphile for shedding light onto this maths problem
Ok, listen-
This is so interesting ...probably need a second part ....
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...
If my physics teacher watch this guy drawing the angle of incidence from the surface but not the normal line, he would cry..
Vincent Han i was thinking the same
not sure how I ended up watching this, but I saw the whole video. So interesting, very smart guy.
Great video-such an engaging dude.
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!
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%.
So, they are illuminating the illumination problem! Great!
I found this video very
*puts on sunglasses*
illuminating
YEEEEAAAHHH
Wow
"So triangles of course are convex. Thank you for that observation." :-D made my day
Had my Linear Algebra Lecture with Tokarsky earlier today :)
Fascinating!
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
Episode 4: The Dark Room
h4ppyh4rdc0ril4 that episode was sick
Robbert R the whole game was.
bay over bae
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 was very relaxing, but I did fall asleep
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.
2:34
also called 'boring'
I want to start of by saying I really enjoyed this video. It really put a new perspective into the way light works and how we should attempt to emulate it; especially when it comes to rendering in places such as video games or for scientific research through virtual purposes.
Something flawed that I have noticed in this video and I am not 100% sure how to explain it so please stand by and try to understand where I am coming from. I think you are looking at the issue as if it's as simple as billiard angles; IE large object hitting a flat surface. Whereas you should be looking at it as small rays hitting spaces between large objects; rather individual particles hitting individual spaces in, around, and between individual atoms. No surface is perfectly flat with no imperfections. All colors of light on spectrum of light bounce at different rates, different surfaces produce different effects on the light that touches it. To assume that the light will bounce at the same angle that it hit at from a large scale point is missing out on the fact that those little gaps greatly change the direction of those rays. We are also assuming that light cannot penetrate a solid object when in fact different forms of light penetrate different objects as well.
My proposal, start with different colors of light on the same rendering style you've provided and once you've gotten to a point where 3 different colors, or spectrum of light observe and reflect different move onto something else. Move onto making slight imperfections, different materials behave differently when interacting, more realistic by proper gaping between atoms, etc, etc. I think this will have a profound difference. Whether or not this will help is a whole different subject.
After the light had enough (an infinate amount of) time to travel though the room and reached a kind of steady-state,
is it possible that one part of the room is brighter than another part?
I mean that an area (not single points) is reached by fewer photons per time unit than another area in the room.
I'm asking, because I got the impression that at 4:50 the left part of the room contains a higher density of light fronts than the right part of the room. Or can this only be while the steady-state has not been reached yet?
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.
This was very interesting, but now I'm wondering about the illumination problem in three dimensions. I would imagine that it would be significantly more difficult to find or construct an example of dark patches/points. But maybe not? If so, that would be a nice follow up video!
I finally know what a convex room is! Many years after failing to understand exactly when the Marathon level editor wanted of me
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"
I know it's the same thing, but shouldn't angles of incidence and reflection be taken from the normal to the surface?
Is only the same in straight surface if you get a curve surface you need to use the normal
You don't need to use the normal, the tangent works the same ^^ and that's what he actually used. Of course on a flat surface the tangent is completely parallel / identical to the surface. But on a curved surface the tangent would just hit the surface in a single point.
It's actually pretty hard to determine the normal on a curved surface as it just sticks out in the wild. Usually the tangent can be determined easier.
How many polygons with curved surfaces do you know?
Who said that we only talk about polygons? ^^ Actually Penrose used an ellipse. Also polygons in elementary geometry are usually described as a "figure that is bounded by a finite chain of straight line segments". However a more general definition of polygon can also use curves / arcs to connect a finite amount of corners.
Not if it's "the same thing."
No candle can light up a room like this guy can
I used to doodle this exact same thing when I was a child, only I didn't know there was maths behind it.