Who cares about topology? (Inscribed rectangle problem)
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- čas přidán 3. 11. 2016
- An unsolved conjecture, and a clever topological solution to a similar question.
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Home page: www.3blue1brown.com/
This video is based on a proof from H. Vaughan, 1977. To learn more, take a look at this survey:
pure.mpg.de/rest/items/item_3...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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In May 2020, it was proved (arXiv:2005.09193) that if your curve is smooth as well as continuous, then you can not only find a rectangle, but in fact you can find a rectangle of ANY proportions you'd like - that is, given any ratio r, one can find an inscribed rectangle whose side lengths have ratio r.
So this means that you can find a square? Because a square is just a rectangle where the ratio of sides is 1:1.
@@teslapenguin1 Yes, though the case of squares has been known since 1944 (Emch proved it for convex smooth curves in 1916). This version of the problem is (relatively) easy because you can apply calculus. The trouble is extending this to non-smooth curves - those that are not infinitely differentiable - where calculus is not applicable, and thus the proofs fall apart.
Hey! Thanks for coming all the way back here and telling us!
Thank you for this information ♥️
Thank you for letting us know! 😁
the hours of work this took.... I can only imagine.
Do you have a link?
Only here to get a notification for a link.
And he's paid well now through Patreon to do it full time. It's a dream job for him, I'm sure.
Victor Hugo bb
Came to the comments to ask what program he uses for these amazing animations and this is the top comment
as he said "isn't that awesome?", when pairs of unordered points are folded into a Mobius strip, I screamed "Hell yeah!". it's just fascinating man :D
I thought: wow that's some creepypasta right there 😂
that mobius strip reveal was a hell of a chekov's gun
I thought, 'thats cool' why simultaneously not understanding at all
türk bulmak nedense garip geldi ne okuyorsun acaba
@@komrad7642 matematik, dümdüz matematik 😄
10:38 I laughed at this harder than I should have just because I wasn't expecting it. The specific intonation he used to say "Not helpful!" just captures the feel of the moment.
Ikr xD
*angry pi*
I lol’d
bro me too it was funny as
\😡/
When you grab a mug and instantly think about a donut, you are either into topology or have a sweet tooth. Ok... maybe both.
Zoltán Pósfai solid joke, nose exhale I will
Me personally, I can only remember the cop stereotype
nerd
@@justaderrickrosestan both his name and his comment are very....Slovakian ;)
ray Williams indeed
It's totally mindblowing how you can talk about things like continous functions/Homeomorphisms, Product topologies, Quotient topologies and ... - is that a commutative diagramm at 14:03?- and make it understandable and sensefull and usefull and everything even to people who never heard of topology. Thanks for everything!
And your patreon accounts shows that by leaping over the 2000$ dollar goal. Congratulations
How would you know he even reads these comments? Since not a single reply is made to anyone.
ProCactus
I only looked for 10 seconds, but on only that video I found at least two comments from himself, and there are much more. Also, even if he did not answer he may still read the comments. And last, even commetns like mine are not only meant for 3Blue1Brown but for everyone reading them
Raphael Schmidpeter
You see reality, Fair enough.
Well, in the end he just concludes by saying "believe me that this strip will self-intersect", without proving anything.
I watched this video back in May 2017, just after securing a place for a Maths Master's degree. I loved it at the time, and made me sure that I'd chosen the right degree. 4 years later, with degree in hand, I still love this video, and is all the more impressive to me now. It's fantastic how the reparameterisation of the space of unordered pairs of points is explained and visualised, without using any of the terminology like R/Z that (I now understand) would be so tempting for an experienced mathematician like Grant to use in an offhand way, but would have been lost on my younger self - add too much of this terminology and I would have missed the beauty of the proof in the first place. Bravo Grant, keep being amazing!
I'm binging the entire channel as I just go into mine next month. Really making me remember why I love this subject.
I'm only in upper secondary school, just found this channel and I'm also loving these videos! I really hope I can come back in 3+ years while studying math in university, then coming back 4+ years later with a degree and nostalgisize. It's like I see people in the comments who have made that journey and I'm only starting, it feels almost surreal
Watching this now again while learning topology formally for the first time (I'm physicist and decided it's time). It all becomes so much clearer. "oh he means homotopy", "this is an equivalence relation", "hey I know, this is a torus!", etc. It was so much fun!
I lost a friend recently because of how much he disliked maths. It was his hatred of topology that really torus apart. (In all seriousness, I LOVE topology)
Hwd405 Yeah I was open to the relationship, but he was closed
Really, no need to be so indiscrete.
I guess you two haven't simply connected.
Hwd405 unfortunately it looks like your friendship is no longer continuous.
Donut despair friendo
There are a lot of awesome videos on CZcams, but this one is probably the video of the year for me.
AMAZING, AMAZING! Super interesting problem, excellent explanation, awesome animations. Also happens to be just the right level for me (not too easy, not too hard). So thanks for this amazing work!
I'm really glad you enjoyed it! I think I said this in the video, but it's one of my all-time favorite pieces of math.
How about a short series on all the millennium problems, just explaining them in the awesome way you do? That would literally be heaven..
3Blue1Brown why you stopped working with Sal
+Ashraya Indrakanti What?
It amazes me how you can make such complex topics so easily understandable to anyone who is willing to think for a few minutes. Definitely one of my favourite videos ever!
These videos are certainly not for the unwilling and impatient. You just have to be curious. Glad you got something out of it too.
This demonstrates the capability of the average human. Many are often afraid of delving into Math because they feel they "aren't cut out for it", or have never made an honest effort.
I believe the truth is that *anyone* can learn advanced mathematics so long as it piques their curiosity.
WOW !! so clear !! I believe these type of visualization are grasped in the minds of mathmatitians but never revealed during demonstrations and articles look so abstracts and inaccessible requiring long study and backgrounds.
This is crazily smart but then does a circle have infinit intersections on the top? How does it work with the loop and does the amount of Intersections have to be odd?
I am a bit confused. This is very hard for me to understand. Is this for pregraduates and higher levels only? I am almost 16 and I struggle to follow, nevermind understand
I’m about to take algebraic topology. Since we’re stuck at home due covid, my topology professor is uploading some of his lectures. I’ve watched some videos already and this videos helped me out a lot to visualise what an homotopy is!
14:36 aaand the music starts playing just as the solution becomes beautiful. It's perfect. This video is perfect.
Yeah it was seriously more of an effective "oh shit" moment than like 99% of Hollywood movies trying to convey that feeling.
No no no no no, if you regularly watch 3Blue1Brown, then if you don't understand something when this music starts playing, you _know_ you're screwed
When I heard it I thought “not the end already! More more!”
Does anyone know the name of this particular song? It’s so elegant but I can’t find it where the rest of the music he uses is :(
@@myrus5722 I've always wondered that too! I just assumed he made it himself (or someone was hired to make it)
I have topology in 2 years and you just hyped the fuck out of me.
Socialist Squirtle Hello fellow math major.
Topology may not be existing, but it is certainly unique
I thought Linear Algebra was the most abstract thing n the world, until I saw it was quite the opposite. Maybe they only teach it that way.
Taras Pokalchuk Everything seems abstract until you're used to applying it
Plus I think a lot of professors aren't very used to applying it, or at least their research doesn't necessarily call for them to know the inner workings of that branch of mathematics.
As a mathematian, what really amazes me is the way you show those unbelievably complicated arguements so simply and ellegantly.
Rarely have I felt compelled to comment on a video. I've seen every 3B1B video at least once (including this one) but for whatever reason, CZcams suggested I watch this again and this video stuck out. This is one of the most clear visual and verbal explanations of a complicated topic I have ever seen. Amazing and fascinating. Beautiful video.
TL;DR : this has not practical use for me. thats not the point, dont care and still liked video. math skills leveled up.
So... I had this math teacher. He said to us "the majority of these things (refering in specific to imaginary numbers) that Im teaching to you are worthless. You will use some of them, and is possible that some of the most obscure math subjects would be used in some weird, special job. But you, in your life, will not be using it." and the class agreed. He told us that he didnt want to teach these things and if everyone agrees, he would just pass the subject. I raised my hand and said no. I wanted to know about. He didnt teach more of that. It was worthless for him and for everyone in that class except me. I moved on until I saw a video explaining imaginary numbers (it was one of the first videos in english I saw. I dont speak english very well, STILL LEARNING :C ). I loved it. Suddenly (well...a month or two later to be realistic) I was subscribed to some science-esque channels like veritasium or vsauce. Look, I suck at maths, science and shit. Im an ilustrator, but you dont have to be Einstein to learn something out of your comfort zone if you like it. I like maths. I suck at it, but I cant live without it. This doesnt have any practical uses for me. But I dont need one. With every video that I have saw, I learn. I understand the universe around me a little more. Thank you, for everyone who makes these videos, for the ones sharing it and for you, to read this comment. Have a nice day.
Well look at that, you're writing in perfect english! And yeah, math is awesome :)
Having curiosity is wonderful. We, Internet nerds, know that feeling of yours.
Damn I feel bad for the people in your class who didn't care about imaginary numbers. No hate but if I was them I'd be kinda tilted
Isn't he required to teach what is on the curriculum? I'd say report him. Anyways imaginary numbers are indeed used practically, for example in electronics
Imaginary numbers are used all the time in many, many STEM field careers. To say they're useless sounds like they were probably in College Algebra or a high school math class and had to merely endure it rather than learn it. Imaginary numbers are insanely useful as soon as you get past the introductory math and science courses.
10:38 "Not helpful" - brown pi
I dunno why i laughed hard
Beautiful video
Fungsi olah raga dari olah rasa fungsi dari olah cipta.
@Orion D. Hunter if you repeat 9 an infinite amount of times, it is equivalent
I scrolled down into the comments right after seeing that
we can make a meme out of this!
Summarizes dead ends for the train of thought following a misleading silver lining to solve a problem, in my view
Oh my, this is my new favorite math video! It's an absolute masterpiece. It's just so awesome that these crazy shapes like the Mobius strip and torus helped us solve a concrete problem. I haven't formally learned any topology yet, but this video has made me excited to learn about it. Thank you so much!
The synergetic sum of all your skills considered, you‘re without a doubt among the most gifted math teachers who ever walked this earth.
Greetings from Germany.
🤨
when see those solutions for finding a rectangle in any shape, I understand the method used but its amazing to think that someone actually came up with those methods.. its hard to comprehend how someone thinks of such things
Math requieres as much creativity as any art
Drugs.
@@sebastiangudino9377 not art, science.
@@eleSDSU well i would say that science involves a lot of art it self, in order to make an accurate conjecture on how the world is, you need to have a lot of imagination
All starts with "What if...?"
Please consider doing an 'Essence of Topology' series..!
Sup
Too much...
Essence of topology. Donut = Coffee Cup.
This would be amazing
Yeah i would love to see such a thing
yeah please.....please...
This 16 minute video probably took me 40 minutes, but worth it. I’m not advanced in a lot of things, but I am curious. I think topology is my new rabbit hole. I very much appreciate all the effort you put into making these videos so concise. Thank you!
I'm a materials science PhD student. There was a senior in my lab who did topological insulator research, I always been interested to her project and asked her about what topology is and got the answer of mug and donut. I honestly love this video so much. It made me realize the beauty of math and getting to know the topology more than just mugs and donuts. Thank you.
You have no idea how much I'm looking forward to seeing this channel develop over the coming years. I'm glad you took your leap into doing this full time and will try to contribute through patreon, as I'm sure many others will too. Keep up the amazing work!
I'm a topologist and I think it's briliant.
It must be awesome to play with doughnuts and rubber sheets all day !
Am i right to say he forgot to make sure the red line ended up correctly oriented when cutting and then making the möbius strip? It seems like it does, but is not mentioned although i think its somewhat important thing.
I not a topologist & I think it's brilliant!
@@DahlenOfficial Check again, but he does not stick the red line onto itself.
Ｉ ｇｏｎｎａ ｓａｙ ｗｏｗ
Let me say this, I had ignored this YT recomendation for a long time and I must say it's the most beautiful video I've ever seen, particularly considering now I fully understand what you are doing whereas when I ignored it I didn't know. The beauty of a random choice.
After watching every video of yours...I wonder how can someone be this good at maths...
You've got some really good abilities of visualising things !!!!!!
"you must know order before you can ignore it" want that on a coffee mug or tee shirt
or a donut
wait
It would be ironical to print it on a coffee mug!!!
1:57, I can look at that animation for hours.
Set it as a live wallpaper!
14:26, even better
Crazy how a free CZcams video can be more educational than some expensive college lectures.
Its not free actually as he doesn't put ads on his channel. So his patreons are paying him to keep making these quality videos. Its free in a sense, but also not free.
Nah, it aint, stupid systems got standartised...
agreed! college...
1. costing you shitloads of money
2. overloads you with a bunch of incoherent garbage that won’t be useful
@@RandomDucc-sj8pd what about this video, then? it's completely pointless knowledge for almost everybody. It's just interesting. college can be useful, but you have obviously you have to be going for the occupations the courses are designed toward.
@@poudink5791 I don't see it as pointless. of course, you need to watch other related videos and have some basics (that you can learn from other videos) about whatever you are interested in or want to use.
by the way, you can find videos for any academic subject you can think of. maths, languages, music, history, architecture, economics, ...
evidently, this video alone won't teach everything you need to know about anything. but it's a piece of knowledge that you can use combined with other pieces to build greatness.
who knows, maybe you can use the knowledge on this video to solve one of the millennium prize problems. would that be useful? sure. it's one million dollars and international acknowledgement.
no knowledge is ever pointless.
I can't remember anyone who spoke in such a perfect way without looking at who is listening. To me, it seems you feel the listener's mind completely. Of course, the topic itself is a different level of pleasure. Extra thanks for those pauses!
You just managed to impress somebody who has written a thesis on moduli spaces. I'm ashamed that I didn't know the Möbius strip is the quotient of the torus by the symmetric action. Great video!
yes, i understand a one of these words.
As a physics student, I'm happy to had understand what you've said (although I didn't realized)
The little pi creatures are SO CUTE
erikjoy12 10:38
@@lucca7716 Not Helpful
We want a film about them.
Just as the minions got theirs
PItures
10:37
This could be turned into a good meme
Easily
Your videos are brilliant in every aspect. They teach, create interest to learn more on a topic and also entertain in a perfect way. I also enjoy the aestetics of your Illustrations a lot. Thanks for this series, I really appreciate!
THAT WAS SO BEAUTIFUL!
This was beautiful, 3Blue1Brown. Simply beautiful.
10:06 answer: unordered donut
Terrific, terrific, terrific indeed! This is the first lecture ever in topology that I understood (leave apart making topology an interesting subject to me, this far I remained totally clueless on things in topology). This is an invaluable lecture, thanks!
I love the thumbnail!
Who cares about topology? "Unsolved".
Hahaha that's great.
Only a mathematician can say that "proving that on any loop there is always 4 points such that these 4 points form a rectangle" is a concrete problem! :-)
Tom Tom my thoughts exactly.
I don't get it...
@@Jupiterninja95 He means to the layman this is far from a concrete problem
This is a concrete problem. I think you are unimaginative. If you think this is not concrete, I wonder what you would think about actual math problems in advanced courses.
Like somebody will rarely ask these obscure questions. But when they do, they go on Google and voila! It has already been solved :-)
Obscure is not the same as not concrete.
The connection between the inscribed rectangle problem and the mobius strip is just insanely beautiful. Though the idea is understandable, it is amazingly ingenius and also explains the utility of studying shapes such as a torus and mobius strip. Really elegant
What amazes the most in 3Blue1Brown is that he's able to construct things in a way that I can (almost) always figure myself the next step. This is didactical genius at its finest
This was the first video I saw in your channel, now a couple of months later I am literately waiting each 3rd friday of the month for your content. Keep the good work, your videos are simple, well explained and compleately intuitive.
_"The Möbius strip is to pairs of unordered points (on the loop) what the 2 axis plane is to pairs of real numbers"_
Really beautiful.
Every video you do my mind opens a bit :)
You might want to see how close that comes to the traveling salesman problem. I wrote a program for the TSP which shrink wraps it using the convex whole method if you put a rubber band around the outside of all the points and then you add points to the loop based on the least distance added first you got a pretty good approximation of the traveling salesman problem. But like your problem here, you can also check to see if any two points overlap. If they do you can reverse the connection and shorten the path, you can also look for higher order differences were three connects have to be moved etc.
You should have some fun with a traveling salesman problem it looks really neat when it's being solved.
Never imagined maths could give me goosebumps! It was so beautiful.
You, sal khan, numberphile etc are the reason why I decided to major in mathematics! I was a guy who used to hate mathematics till the 12th grade/high school but you all have changed that! Very grateful that I have access to all these for free!this video blew my mind, so have others like the essence of linear algebra! I want to say a big thank you! Love from Nepal!
Excellent video! I am eagerly looking forward to the calculus series. One minor suggestion: When you are presenting a beautiful proof such as this, it would be great to cite the original author. In this case, I see that some websites cite H. Vaughan (1977) as the source.
Great point! Adding to description now.
I have a few question if you would be kind to respond.
I wonder if this foundation of riding two distinct points to rectangle can any how be used to explain space time equations ? Or does it have any relation to any of what have been discussed here?
-Is this some what equal to imagining a Fourier transform where you have only a few dimensions and you need to know if the other one corresponds to 3D space and some how evaluate how the shape would be ,but without considering Fourier?
-If you could let me know the application of these shapes starting with the advantage of finding a rectangle?
-if a rectangle with a common center in every moment in time could be framed , how do we derive the equations for this, and if so, how do we use limits and functions?
-If such an equation could be derived , what if a change in external factor day a variable x dash has been introduced, how will it affect the common center point?
Finally,
Is there a simple graphical representation to understand how , Fourier transform is important and why dimensions form?
Expecting a response , please do leave a comment if something has to be clarified.
You got your spark, I hope to get mine.
Wow, I remember watching this video right when it was released, and now I look back on this problem after it was just proven recently, fascinating to see modern mathematic discoveries being made every day
I love watching these videos. I’m a CS + Math major in college and I feel like so much of these videos have given me special intuition in problems making me more successful in my classes
There have been a lot of creators that I've thought about supporting but always managed to rationalize my way out of contributing to by telling myself I can't afford it. You are the first person to really make me change that, because I simply cannot ignore the debt I owe to you for how much you have helped me understand mathematics. I wish I could give you $2^8 per video, I really do! You're going to make it in this endeavor Grant, I know you will. You have to! We need more people like you in the world. Thank you so much for everything!
+TheAgentJesus Wow, what an incredible compliment, thank you so much. Also, great username.
Of course, you absolutely deserve it. I can't wait for all of the stuff that you have planned, and feel incredibly lucky that I get to use your videos to supplement my education. You're the kind of educator that I aspire to be.
Oh, and thanks! Now if I could only remember what compelled me to make it haha
This is an outstanding use of graphics, video, and speaking to make an exciting connection from abstract concepts to something clear. I hope there’s a follow up due to recent work on this problem!
Math is one of the beautifull things in this life. Strange how I lost so much of my interest in novels and philosophy while my interest in math grows... Math just feels more pure an honest.
the aesthetics of math is just so beautiful, i miss it so much in humanities classes. the kind of logical, elegant solution to things that are, after solved, obvious. Still, as many great philosophers were mathematicians, you may think of it as a stimuli to the mind, and one day go back to philosophy more knowledgeable.
As time goes.on, my interest in philosophy wanes as it is replaced with a love of the pure beauty and rigor of mathematics.
"You can always find a rectangle, so long as you consider any pair of points to be a rectangle."
-3blue1brown, 2016
"NOT HELPFUL!!"
-3blue1brown, 2016
Rectangle of infinitely small area
Terence Tao recently published a result on this unsolved problem, the paper is: "An integration approach to the Toeplitz square peg problem" on Arxiv.
What was the result?
"We show that the answer is affirmative if the curve consists of two Lipschitz graphs of constant less than 1 using an integration by parts technique, and give some related problems which look more tractable."
This video is so intellectually (and intuitively) satisfying. It reveals such deep mystery in how we can look at the world around us. Thanks so much for doing this is such a well-designed graphical output. Really, really outstanding.
"When I was a kid" 0:20
This animation is the cutest thing ever
Markus Persson supported you on Patreon? The indie developer of Minecraft? That's awesome, lol
Btw, this is the best math video on CZcams. Not sure how exactly to explain why, but it _is_ the best.
No other channels match the genius intermingling of intuition and computer graphics to make it entertaining and approachable like this channel.
How long it must take to make these...
Can the same problem be proved for a loop in 3D?
Essentially what I'm asking is, is there a natural extension to this problem in nD?
i like dis question.
Since möbius strip in 3D doesn't (have to) intersect itself, you're not granted even a rectangle in 3D closed curves. Edge of a common möbius strip is the example.
Other possible extensions would be trying to find a cube or a prism on closed 3D surface. But I think there's too many degrees of freedom in 3D that finding a counterexample wouldn't be a problem.
I don't see a problem with extending this to curves in R^3. For curve S, map each pair of points A and B to (xm, ym , zm + d) where (xm, ym, zm) is the midpoint of the segment AB and d is its length.
I don't understand kasuha's point. If the mobius strip doesn't have to intersect itself when being mapped to an object in R^3, the proof in the video would not make sense.
The edge of the common mobius strip has infinitely many inscribed rectangles, although I suspect the edge can be carefully arranged to eliminate all but 1 of them.
@DrGerbils The problem with immersing Mobius strip as you described is that, since the points of the curve are not on the xy-plane any more, the point-pairs with different distances can yield the same point. (ex: the pairs (2,0,0),(-2,0,0) and (0,1,2),(0,-1,2) would both give (0,0,4) with your description, even though they neither share a common midpoint nor are equidistant, hence don't form a rectangle).
The contradiction in the proof given in the video doesn't arise from the assumption that Mobius band cannot be embedded into 3-space (which is false, it can be embedded into the 3-space), but from the fact that this cannot be done in the particular way described: sending the whole boundary circle to xy-plane and the interior to the upper-half space z>0. The reason this cannot be done is that, if this were possible, then we would be able to attach a disk to the bottom of the resulting surface and obtain an embedding of the projective plane to the 3-space, which is known to be impossible.
adakarga Thank you for the counter example. Well, back to the drawing board.
Wow! I'm in awe. I'm reading "The Universe Speaks in Numbers" and got to the part where they are talking about topology. Since the book is more about the history of physics and mathematics and the people involved, it doesn't go into a lot of detail. But I wanted to understand what topology was. This was one of the videos I watched in my study and by far the most intellectually challenging and stimulating.
I wish geometry in public school touched on this a little. I was a math heavy kid and knew I liked science, but never found any topics between those two very interesting in high school. it's just simply what I was best at, but not only is this super interesting but it also helps my overall understanding of math and geometry and why we have the laws of geometry that we do and why they apply to our dimension in those ways
Really beatiful proof!, and really amazing that the representation of a pair of unnorderd pairs in a close loop is a möbius strip!
P.S.: Did you noticed that Markus Persson (aka Notch, the creator of Minecraft) supported you in Patreon????? That's really amazing!!!
Unbelievable how you make things easy to see. Thank you for your work.
I wonder, are there any algorithms that can find a rectangle quickly in a loop?
Also looking back on your videos, it’s amazing how the quality just got better and better. This is also one my favorite pieces of math.
I saw the thumbnail for this video numerous times, saving it for a “rainy day” . After I saw the new 2020 results, the picture immediately came to mind - I think the Quanta article used a similar graphic.
Brilliant and inspiring as always. Can't wait for that "essence of calculus", although I personally already consider myself to understand the essence and would love to see one about topology, which is mostly gibberish to me.
By the way, thanks for releasing the source code for manim, I've always wondered how you make those magnificent animations, and I hope you don't get too frustrated writing the code for your animations. Wish I'd manage to run it on my Windows 10, but I still haven't gave up patching!
Rock on, 3B1B
This is amazing. You know how to hit the intuition directly.
Hi Grant. Thanks for your generosity of knowledge and clarity in sharing it. I’m just beginning to grasp some of the rules the behind the beauty and the beauty in the rules of math in more depth, as an artist with a consistent string of math misdemeanours on my long-distant high school record, having avoided the felony of diplomatic failure by the skin of my teeth. Please pardon the ignorance in my probable misuse of forthcoming terms and tenuous grasp of concepts from math and physics. I’m interested in whether, in your animation, if you were to stabilize the inscribed square at the centre of the dynamic diagram, so the closed loop moves around it, is it correct to think of the differential iterations of the loop as the possibility space or phase space of the inscribed square, and the shape of the surrounding scribble as its attractor, for that given loop topology? I’ve been introduced to these concepts through digging into dynamical systems / chaotics a little, but understanding how an attractor is produced has eluded me until thinking about inverting your animation. I’d love to see such an animation, as I think it will help visualize the inscribed square as a metastable continuity of form (subject to the variation in the structure of the loop) and an emergent (square) form of organized complexity amid the chaotic scribble of the loop. I also wonder if we were to take this approach of centring the square, and average over the shape of iterations for any given loop structure, would there be a homeomorphism across all averaged loops, creating a generic attractor for square-shaped metastability/metastable squareness (if this is a correct application of ‘attractor’)? A final question in the string: please can you advise what software you use to create and animate your diagrams and if you can recommend any basic, non-specialist versions with a gentle learning curve? Cheers, Jem
Thank you, I am a high school student, and I am thankful for you to let me take a glance of topology. I didn't understand any of it until I watched your video. It hit me so hard just like a dopamine release. -Jan 2023
I must say thank you for showing such an elegant solution to an otherwise difficult problem. While not a mathematician by trade (I am actually a physicist), I do have a deep appreciate for mathematics and thoroughly enjoy demonstrations and problems such as these. I am truly grateful that you brought this problem to my attention since it will provide me with quite a few days of entertainment trying to see how far I can get in proving (and then failing to prove) the inscribed square problem. Cheers.
At the beginning of the video, you say, "...squiggle some line *through space* in a potentially crazy way and end up back where you started," but then the rest of the video (and the inscribed square problem itself) concerns itself with *plane* curves (emphasis mine). The first time I watched the video, I came away with the impression that this property applied to any curve in *3D* space. Watching it again clarified that the inscribed square problem really applies to plane curves. This is the only serious error in the video. Other than that, it is extremely well done. Thank you for making it!
After watching the video, I tried to came up with a solution, which ofc was not successful, but gave me some maybe-good ideas:
1. For every point of a 2D loop, there are infinitely many deltoids inscribed into the loop which contain that point.
2. There is at least one inscribed rhomb for every point of the loop, so that the point is a vertex of the rhomb.
3. If one can show that by continously moving our arbitrary point around the loop, the rhombs undergo a continuous transformation, then there must exist an inscribed square.
Proof of 1:
For each point of the loop, there are at least two points the same distance d (less than the maximum distance) away from that point (can be shown by drawing a small enough circle). Then, draw a straight line from our first point, which is purpendicular to the line connecting the two points mentioned above. It has to pass through another point of the loop! The four points form a deltoid. The same method gives different deltoids for each distance d.
Non-rigorous proof of 2:
Start with an arbitrary point A. Then, consider points of the loop "on the left" of A and "on the right". For every point B "on the left" there is a point C "on the right", so that |AB|=|AC| (see Proof of 1). Let one of the points the furthest from A be "o.t.l." and "o.t.r." at the same time, so that it's the only point except A with this property; let's call it M (from "middle" or "maximum"). OK, here it gets a bit tricky.
Let d_l(B):=|AB|, d_r(C):=|AC|, where the domain of d_l is the set of points "o.t.l." and dom(d_r) is analogous. By moving point B' from A to M, we see that d_l(B') has its local minima (starting with |AA|=0) and maxima (ending with |AM|); same thing with d_r(C').
Now, let's move both B' and C' from A to M, keeping them on respective sides of the loop. Is it possible to do so while preserving d_l(B')=d_r(C')? Yes :) bc each time we hit a local maximum on one side, we can proceed further by going back on the other. Then, when we get to a local minimum on the side where there was a local maximum before, we again switch directions on the other side. We go further and further by kind-of-zigzagin, having d_l(B')=d_r(C'), until we get from A to M "o.t.l." and "o.t.r.".
Why do we need this? It means we can continously change two, and in concequence three of the vertices of an inscribed deltoid with a "constant" vertex A. Now, let's focus on the segment B'C' - it moved from "near A" to "far from A", which means it must have been the same distance from A and from the 4th vertex of the deltoid at least once; somewhere there we got an inscribed rhomb.
Proof of 3:
Imagine an inscribed rhomb ABDC, |AD|≤|BC|. Imagine we transform the rhomb continously, while changing one vertex from A to B, where we get the same rhomb we started with. Since the width and the height of the rhomb changed places, somewhere along the process they were equal. Yes, that means there was an inscribed square :)
My knowledge and skill are too little to even try to proove the thing needed in (3.).
If u can proove or disproove it, then be sure to reply to my comment :))
I just remembered what made me love topology. Those two videos that talk about turning a sphere inside out
Awesome! I love when math material gets me new ideas!
Question:
Imagine the intersection of the mobious strip when folded onto the 2D plane.
That intersection should not be a single point.
The intersection should therefore be a line.
The ends both represents different sets of points with 0 distance.
They have a commen top-point.
(Actual question)
Would it always be possible to pick 2 un-ordered points from that line, which would form a square?
Perhaps it is possible to make and visualize an expression of the formed rectangles side lengths.
And then find a commen side length of those rectangle point pairs.
Man!Your videos are really beautiful!I loved how easily your explanations and animations helped me understand this seemingly difficult topic(I'm actually in 10th grade,so I know nothing abt Topology,but still I understood it every bit!).
It continuosly amazes me, the ingenuity of these constructions. Kudos to you on introducing them to the masses, they are worth knowing.
Re: inscribed square problem
As always, very nice explanation with equally pleasing graphics.
Once your find your inscribed rectangle with points (a,b) and (c,d) with ac = db, could you not extend your boundary conditions on your surface to also require ac = ad and not equal 0 to avoid "point squares?" (or bc = bd or ac = bc or ad = bd) Basically require it to be a square.
No, not at all. That'd require a completely different proof or disproof.
You would have to define a different function that also captured the information about ac = db. You _could_ try to do it the naive way, getting a 2d surface on some sort of 4d space. But due to the mobius trip still representing the unordered pairs or points, you could actually map the mobius trip into this new 4d surface without it intersecting itself, due to how braids work in 4d.
Search "numberphile braids 4d" to see what I mean.
This proof gave me what I can only describe as a spiritual satisfaction with mathematics. It continues to be the one and only thing that can inspire a childlike sense of wonder in me. I plan on taking a little out of my paycheck each week to support you through Patreon after this one, sir.
Sir, I found this video really interesting, thank you so much for creating it! I've decided to explore this problem through a hands-on experimental investigation. Could you please suggest any methods or procedures that I can follow?
Thanks for showing the insightful and visual aspects of maths, often neglected and that I feel contain the beauty and profound meaning of maths.
I've seen increasing interest in topology for Materials Science. This just confirms the transversality of concepts.
Regards
My first reaction was like "meeh..topology". Numberphile and other channels are doing a lot of videos about that topic and i never was really interested. But this proof you just show us blew my mind. And now I can't wait to see more of this!
My mind is blown and when I have glued it back together I need to rewatch this video .
These are really accessible and clear videos which make very abstract concepts much clearer. Thank you for the hard work!
I think it would be nice to show the curve traced by points of type (X, X) on the torus itself and correspondingly, might also be good to give an idea about how a toridal ring shrinks to a Mobius strip when order is remove.
Nobody:
3b1b: Consider the Mobius Strip as the orbifold of the torus.
This is so over used
How about the intake manifold of a Taurus?
3b1b? Is that a new anarchy Minecraft server?
@@LostSwiftpaw SciCraft on mathematical steroids
@@simo4875
Are you assuming Scicraft isnt already on mathematical steroids?
Whoaaaa...that was a fucking beaitiful solution, my fuckin god. Well worth the wait.
You know you are doing gods work when ALL your comments on your youtube videos are other humans describing how beautiful and satisfyingly perfect what you have created is.
This is the most beautiful video you have made in my opinion.
Last minute and a half, when the music starts... Massive smile just appeared on my face. :)
18 minute 3Blue1Brown video!? AWESOME!! :)
Ok, but but to my point. That was amazing! As with most of your other videos, this motivates me to learn more on topology. Also can't wait for essence of calculus to come out. If you read this comment 3Blue1Brown, do you know approximately how many videos will be in it? Will it be out by January? It would make an amazing Christmas present, but it would probably be premature to release it then. I don't know.
This comment is getting lengthy. YOUR AWESOME!
EoC will have 10 videos (plus whatever scope creep happens), and I'm hoping to make one every other week. So...20 weeks from now?
Awesome! I'm super excited. :)
You literally have just convinced me to learn topology. Man, the list is getting bigger and bigger!
Topology is awesome in that it allows us to get back a geometric notion of relations that would "only" be represented by numbers in conventional algebra.. i am just at the beginning of my learning of abstract algebra, but definitely have to dive deeper into these topics!
Who else is here upon reading about Greene and Lobb's 2020 result?
Me
Moi
Yo
Me
Can you link?
holy shit, when you cut the triangle and realised what you were about to do I had a mathgasm, this tops the Gaussian integral for me
So if a circle has infinite inscribed squares, then the 3d plot of the circle should intersect itself an infinite number of times.....shit.
There's certainly a crumpled nipple at the top of the breast-like curve that forms over the circle. ;P
Technically no as all have the same midpoint and length of all diagonals are same. So it intersects at just one point but that single point corresponds to infinitely many solutions.
Actually if you watch an animation of unfolding a moebius strip into a circle it just does that : in intersects itself infinite times
I think intuitively the Mobius strip makes the most sense. A Mobius strip is a 1-dimensional surface, the action of cutting/gluing a 2d surface would also mean the pairs of points on the 2d surface map to points on the 1d surface.
I think the Mobius strip is a very elegant and intuitive solution to the 2d unordered pairs of points.
Of course, that leads me into thinking about triplet points, 4d points, 5d points, etc, and their corresponding surfaces.
This is one of my favorite videos of all time. I come back to it often.
I really don't have words to thanks. You have made the concepts so simple and visual ! I think all the viewers do acknowledge the time effort you gave. Its awesome sir. Keep up your good job. Good going :)