This open problem taught me what topology is

[@DrTrefor]

That feeling when you did your PhD in algebraic topology and then watch a 3blue1brown video and think "oh THAT's what topology is!" :D

[@BadChess56]

Yeah...

[@display9138]

Merry christmas

[@codahighland]

So can you answer the question implied in the video about what practical application this problem might have? What might employ this as a step within a larger proof, perhaps?

[@actualBIAS]

You both should do a lecture together. 1.5 hours. On a strangely interesting topic.

[@139-b7j]

Obviously. Algebra is fake math.

[@carykh]

I love the realization at 21:20 thinking, "this isn't going to work... wait, you could make a Klein bottle!' and then realizing that the most famous property of Klein bottles (the fact they self-intersect) literally solves the problem

[@ThatShushi17-mc7ct]

i always love it when in these videos you suddenly get this eerie feeling of _now where have i heard that before?_ (generally previously in the video, but there are times like this too) before he reveals exactly what you were thinking about. i also knew beforehand that two mobius strips connected end-to-end make a klein bottle, combined with the spoioer at the beginning, i was freaking out a bit when the napkin was pulled back up.

[@suspicioussand]

yo hey carykh

[@ObjectsInMotion]

I had the same exact realization! Any discussion of a Mobius strip in my opinion *begs* for it to be stitched to a Klein bottle!

[@wasp795]

greetings cary

[@proton..]

i love you cary

[@Finkelfunk]

Our math professor once told us that "No matter how well thought out you think your examples are, some topologist is going to come along with some inconceivable 12 dimensional shape and prove to you that you are wrong. They are basically the trolls of the math world." before going on a tangent about how there are so many neat and tidy theorems in mathematics that would work out beautifully if it wasn't for some topologist finding that one exception in some hell child of a construction in the 28th dimension that completely breaks the theorem. I think after this video I finally get why this is the case and what that means to think about the mappings of math problems to specific objects in higher order spaces. Thank you :)

[@gabrielvieira3026]

The category of the topological spaces is a wild universe There's a classic textbook called Counterexamples in Topology, but the counterexamples are a lot more abstract than 12 dimensional spaces

[@Merthalophor]

for one it highlights the necessity of rigorous proves. Things that make sense "if you think about it" aren't always true.

[@bhbr-xb6po]

Analysts just need one dimension. Enjoy your continuous everywhere, differentiable nowhere function

[@elia0162]

@@gabrielvieira3026 AAH YES THE TROLL MASTERGUIDE

[@Alan-zf2tt]

Please excuse my naivety but ... The leap to 4 dimensions was it more a case of not quite orthogonal bunch of four tied variables? So a surface really does seem to be a natural consequence?

[@ashu95tw]

As an example of practical applications of topology, a conference paper published in 2023 has a main component (singularity-free embedding) inspired by the 1st edition of this video. The set of parallelograms are embedded one-to-one into a 6D space without discontinuity, so that an NN can easily learn to estimate the position of a parallelogram from a somewhat deformed image. I am the first author of that paper, "ReMark: Privacy-preserving Fiducial Marker System via Single-pixel Imaging", and I want to express my deepest gratitude to 3B1B.

[@ancientswordrage]

I would love to see this animated to help explain it

[@PinkeySuavo]

i still dont know what practical application is xD

[@IHarleyWin]

@pinkeysuavo it is neural networks.

[@Sarah-lk8ye]

wow, wholesome😊

[@ashu95tw]

​​@@PinkeySuavo I'll try to explain it briefly with a simplified example. Imagine that you are building a reader for a square-shaped 2D barcode, something like QR code. To simplify things, let's ignore perspective distortion and assume we only have to recognize the boundary of a fixed-size square from a photo. For some reason, we decide to train a neural network (NN), whose input is a 2D array of pixels, and outputs 3 numbers - the x, y coordinates of the center of the square, and the rotation angle in the interval [0°, 90°). You will soon notice that the NN produces large errors around 0° and 90°. This is because NNs only approximate continuous functions well. 0.1° and 89.9° look almost identical in the input image, but their outputs differ by a large value. This kind of discontinuity can be resolved by embedding θ∈[0°, 90°) into a higher-dimensional space, a pair of 2 values: cos(4θ), sin(4θ). Now the NN outputs 4 values: x, y, cos(4θ), sin(4θ), and since it became a continuous function, the training converged sooner and the NN produces much less error. Hope this helps in explaining the application of topology. You can find an advanced version with perspective distortion addressed in the paper.

[@DanielKRui]

New editions for old videos is a wonderful wonderful idea!

[@BadChess56]

I KNEW my deja vu was founded!

[@hamiltonianpathondodecahed5236]

I also remember hearing that this problem was solved

[@redstocat5455]

Happy christmas

[@gaelonhays1712]

Agreed!

[@alfadog67]

YES! The animations are top-notch!!

[@santherstat]

part of why I kove your videos is that there are these little moments where I can see what's about to happen. "Oh, we're going to be plotting a shape, and so we have a rectangle where it intersects itself!" You make me feel like I'm part of the discovery, and that's a beautiful feeling

[@HypherNet]

You have turned math pedagogy into art and become a master at it. What a wonderful gift to our species that will likely resonate far longer than any of us will live.

[@benj6964]

The exact thing I wanted to comment, 2 minutes in

[@pizzarickk333]

​@@benj6964This comment articulates well what I have in mind when watching 3b1b.

[@johnsmith1474]

That's a bit much, you mean that you like the pretty moving graphics and think they are like art. Fractals did that back when there was actual LSD available for consumption.

[@crimsnblade8555]

@@johnsmith1474 no, not the pretty moving graphics, art as in the art of teaching

[@HypherNet]

@@johnsmith1474 What I mean is that he is bringing deeper understanding of math to a wide audience. When I say "resonate far longer than any of us will live" I mean that I've learned things from him that really helped me in my pursuits making art using 3d printing and LEDs. I hope some of my work has inspired others as well. No doubt their work will inspire even more. And I am only one of 6 million people here learning. So many cool things in this world will have been made possible or made better, or faster, because of these videos. And that's pretty cool. (to borrow from Chris Boden)

[@Bolpat]

9:44 The maths teacher I had in grade 10 in every exam included one task where there was superfluous information. It's genuinely good because in real life, you almost always have more data than is strictly necessary to solve something.

[@Mepharias]

My physics professor last semester was very good at writing test questions. In theory that sounds like a good thing. In practice this meant the he knew exactly how much information he could withhold while still having a solvable problem. It was awful and me and the other students that were successful have trauma bonded. I'm smarter for it, though. Those questions were fun to bang my head against. Just not at the time.

[@animowany111]

@@Mepharias I have trauma from one particular question that mentioned a length of copper wire in a water tank, which we had to calculate based on how long it took the water to evaporate. There was literally only like one constraint, and somehow that's enough to connect resistance, amperage, wattage, wire length, wire thickness, heat dissipation, and evaporation rate.

[@icantollie]

I had a thermo professor who would do that; like sometimes he would mention the color of the housing of the engine in a problem about Otto cycles versus Diesel cycles. He said he'd throw those in to hammer home that as engineers half the time you're filtering out totally irrelevant information from the actually relevant info given to you by whomever you're doing the work for

[@Heulerado]

@@animowany111 That sounds fun. Can you find the problem? I'd like to try

[@johnchessant3012]

26:45 "the reason that mathematicians get really excited about bizarre properties and impossibilities is not just aesthetic. it's because when you're looking for logical proofs, constraints and impossibilities are your fuel for progress" woah

[@gaelonhays1712]

This is why we play puzzle games and add rules to sword fights and D&D. If you can do everything, there's just chaos. If you're constrained, then you have boundaries to push against, so you can orient yourself; you have obstacles to remove or avoid, so you can specify what tools you need for them; you have an actual goal, and can therefore move towards it. (The Arcadian Wild has a brilliant song called The Storm that lines up well with this. I recommend the song and the band to all.)

[@santerisatama5409]

Mathematics is a dialectical science, as Proclus attested. And we like to do our reductio ad absurdum proofs as thoroughly as we can, by going as deep into the absurd as we can. Yet, for our humanity it is not adivisable to go the route of Frege and Cantor and go crazy in the bad way.

[@santerisatama5409]

@@gaelonhays1712 Deriving theorems from ex falso "axioms" leads to the logical Explosion of mathematical truth that ends up in truth nihilism. That's the core lesson of Formalism of arbitrary language games.

[@gaelonhays1712]

@@santerisatama5409 I'm not following. Explain it like I'm 5.

[@rileybrown342]

​@@gaelonhays1712 Rules define the game. No rules, no objective, nobody cares.

[@janmaris7358]

we should never take Grant's videos for granted... this one is 'next level' in all relevant intellectual dimensions: next level graphic visualization, next level teaching, next level problem solving, next level reasoning, next level comprehensiveness, ....

[@marcoazevedo7885]

I loved my Christmas gift. Everybody would love having a topology class for Christmas

[@S.G.W.Verbeek]

Merry Christmas evening!

[@marcoazevedo7885]

​@@S.G.W.Verbeek But now is 3 pm

[@S.G.W.Verbeek]

​@@marcoazevedo78857 pm here😅😆

[@BadChess56]

It's 1:16 pm christmas eve for me...

[@0x4566_emena]

Some of us want family and money

[@echo5delta286]

Several things you said here remind me of one of my favorite quotes from 2010 Fields Medalist Cedric Vilani: "There has to be some useful and some useless. Sometimes the useful will become useless. Sometimes the useless will become useful. [...] You have to allow some uncertainty in the system. If you try to predict, you lose the most interesting parts."

[@RenderingUser]

The way the videos on this channel activate my neurons is unlike any other channel on KZbin. Each new video only further cements this channel being the best educational channel on KZbin.

[@DancingRain]

I don't think "Bravissimo" is a strong enough word. This is the most beautiful mathematics video I've ever watched. Had this been a lecture in meatspace, I'd be giving you a standing ovation. This reminds me so much of the "higher" mathematics courses I took in high school and college, where it felt like every semester I was handed at least one golden key that unlocked a hundred solutions.

[@3blue1brown]

Oh wow, thanks! Do you have any favorite golden keys that came up in this “higher” courses?

[@TheDeidara2001]

@@3blue1brown have you done a video on Riesz Representation theorem for linear bounded operators? I remember doing functional analysis and with that theorem suddenly the word 'dual' in dual space emerged to me as more proof mathematicians have often very poetic inclinations.

[@FacultyofKhan]

This video has now taught me what topology is.

[@BatSignalJammer]

This comment has now taught me what topology is.

[@zamplify]

I still don't know

[@omlett6482]

@@BatSignalJammer This reply has now taught me what topology is

[@vikramadytia7158]

This video has now taught me that i was wrong about knowing what topology is.

[@pranshukrishna5105]

@@omlett6482 This reply to this reply has now taught me what topology is

[@pranayblue]

The fact that I'm able to understand this video... is blowing my mind. When you glued those edges to form a torus, it just MADE sense

[@TrackpadProductions]

My introduction to topology was that old "Turning A Sphere Inside Out" video. For _years_ I had no idea what they were on about or why anyone cared. I, uh.... I get it now.

[@LeoStaley]

Mathologer did a video on tunrinng a sphere inside out with a newly discovered, more simple way of turning the sphere inside out. I loved it.

[@chimkim]

Mine was the huggbees version. Equally interesting for different reasons

[@elementgermanium]

My first instinct when I saw the mobius strip to circle mapping was “but he pulled a loop tight and that’s cheating!”

[@redstocat5455]

Happy christmas

[@warriorsabe1792]

@@elementgermanium lol me too, I think that was just 3b1b's animation since I can imagine flipping that loop over like if you had a twisted cord

[@MichaelPiz]

When you started morphing the shape into a Klein bottle at 21:21, it completely blew my mind because I was imagining bizarre twists of the shape to get the yellow arrows to point in the same direction so I did _not_ see the Klein bottle coming. Fantastic! I immediately ran and got my Klein bottle, held it up, and stared at it for a good 10 minutes as my brain went BRRRRRRR.

[@pabloalvarezdominguez7222]

The first time I saw your video, I didn’t know what topology was, but it made me decide to study it. It wasn’t easy because I couldn’t understand the connection between what I was learning and what you had taught me. Then I learned about hyperspaces and symmetric products, which I soon realized I already knew something thanks to you, and had one of the most satisfying experiences of my life.

[@redstocat5455]

Happy christmas

[@debrachambers1304]

@@redstocat5455 Merry holidays

[@ver_nick]

@@redstocat5455 Happy christmas!

[@andremouss2536]

The quality of the video animations - not to speak of the quality of the commentaries, of coures - is truly amazing. Anyone should fall in love with geometry, topology, logic watching that wonderful piece of human intelligence. There is no 'thanks' big enough to celebrate this.

[@TheGamer583]

I was fortunate enough to be able to attend a talk by Josh Green on his new results. It is remarkable that such a simple looking piece of math has deep connections to highly abstract ideas, namely those coming from symplectic geometry and topology, Lagrangian flows and intersections, and more. It's refreshing to see this kind of "higher math" still be used very concretely.

[@OMGitshimitis]

You probably know significantly more than me, why can't we use the fractal shapes that present the problem as limits that we approach with increasingly complicated smooth shapes? Is there some dodgy bit of reasoning I'm doing here because in my brain it feels like that means we should at the very least be able to make arbitrarily close shapes to squares inscribed by fractals. But I also feel like that's such dumb reasoning it must be wrong.

[@TheGamer583]

@@OMGitshimitis This is not dumb, and in fact very good intuition. Trying to approximate your non-smooth curves by smooth ones is a great initial thought, specially given that this strategy does work for other problems. For example, the Jordan Curve Theorem says that if you have a continuous curve in the plane without self-intersections, it necessarily separates the plane into an "inside" and an "outside" region. One way to prove this is by approximating your curve by polygons, for which the result is seen to hold more easily, and then working out the limiting details (which are intricate). But in this instance, for the square peg problem, I believe (and memory fails me) the trouble is that your square/rectangle may "collapse" under this limiting process, even if you approximate your non-smooth curve by very nice smooth ones, or even polygons. If you know the result is true for polygonal curves, you know that they inscribe squares, but you don't really know "where" the squares are, and if they are for example decreasing in size or not. Hence the need for extra tools and approaches. This whole discussion comes back to the idea that limiting statements are some of the most important and useful ones in math, namely showing that if some sequence of objects tends to a limiting object, then this limit will have some of the same properties as the ones approaching it.

[@isavenewspapers8890]

⁠@@OMGitshimitis For example, maybe as the curves get more and more jagged, the squares get smaller and smaller. In the limit, all four vertices of the square would be the same point, which we're not allowing.

[@TheGamer583]

@@OMGitshimitis This is not dumb, and is good intuition. This strategy, of proving the desired statement for simpler clases, and then showing that the general case is in a sense built up from the simpler cases (such as by taking limits), does work for many problems. For example, the Jordan Curve theorem states that every continuous closed curve without self-intersections must bound an "inside" region and an "outside" region. This is a remarkably hard thing to prove, and one strategy is to show it's true for polygonal curves and then work out the details in the limit by approximating your curve by polygonal ones. But this still requires you to have good control of these limits, and hints at why it may be harder in the square peg problem. If you approximate your curve by polygonal or smooth ones, you can show that they have inscribed squares. But you have little information about them, their position, sizes, etc., so in the limit they may actually collapse to points, and you cannot guarantee they don't, which shows why you may need different approaches. These themes are prevalent throughout math, and limiting results are very important and useful to guarantee when things like this work out or not..

[@JonBrase]

This simple looking piece of math also has connections to why you don't fall through the floor.

[@hifriend7789]

At first I thought any solution to this problem would just fly over my head, but I was able to follow ever step in the solution development and it all felt very natural. What a great video 👏

[@isomeme]

Your casual mention of the isomorphism of the set of pairs of points on a circle to all possible musical intervals, combined with the material earlier in the episode, exploded my brain. Now I have IDEAS, and my main challenge will be getting any sleep at all tonight. Thank you!!

[@amattenet]

exactly... what does the "you cut a m strip in two and get a longer strip" mean in terms of interval? etc

[@hyperduality2838]

Mobius loops = spinors! The Klein bottle has two Mobius loops -- self intersection or duality. The left handed Mobius loop (left spinor) is dual to the right handed Mobius loop (right spinor) synthesizes the Klein bottle -- the Hegelian dialectic. Vectors (Bosons, light, waves) are dual to spinors (Fermions, particles) -- wave particle duality. Spin up is dual to spin down -- the Dirac equation. "Always two there are" -- Yoda.

[@ВладимирФедосеев-б1у]

This is absolutely crazy work. Starting from working on the script and fitting all this in your head, to making a freaking visualization framework and making such a great visualization

[@Shreyam-q6b]

It always had bothered me that I faced problems regarding basics of differentiation, Integration and Linear algebra. But after I attempted the courses of 'Essence Of Calculus' and 'Essence Of Linear Algebra', now I totally feel confident about my basics and foundation of Calculus and Linear algebra. The course of 'Differential Equation' has also set aa strong base in thinking about ODEs and PDEs more clearly. If I achieve something in my life, this man will get full credit.

[@drgothmania]

I love the way mathematicians keep abstracting and abstracting and finally connect everything altogether in a subtle, beautiful concept. I am an epidemiologist who is interested in measurement error, and has been always thinking about mapping everything I measure onto some different axes or surfaces. Feeling like I need more maths to make sense of what I'm interested in...

[@lethargicastengah572]

Check out measure theory

[@Ancor3]

Can we just take a moment to appreciate the amazing visuals here? This video had some gorgeous animations!

[@Eysc]

its a python lib. that he created, its called Manim, if you're interested its all free and a lot of resources to put it in use, and yes its amazing

[@mchabbblt]

being able to program these clear and accurate visuals shows how deep your understanding goes. thank you for sharing

[@Xeridanus]

He wrote the animation library in python and it's available for anyone to use.

[@hiteshvs9743]

This video is such an amazing way to think about the topological shapes and what they mathematically mean. I truly love how a seemingly irrelevant question can help you dive deep into a subject so abstract and fascinating!

[@perrymaskell3508]

I stumbled upon this channel 4 or 5 years ago when I started an Applied Maths and CS degree. I got my Engineering degree BC (before computers / the Internet), so finding this channel that explains things with such wonderful live graphics was mind blowing. Pleasant voice, and focus on the topic, not the presenters face. The 2nd Edition shows the growth in graphics, presentation and processing power. It is unbelievable that someone can convey such interesting and complex topics in such a clear and easy to follow manner. To go from problem, to understanding, to means of presenting, to final presentation is fantastic. Please keep up the beautiful work. Greatly appreciated!

[@sarbajitmazumdar1905]

Great work! Really good animations and touch of rigorousness.... As someone working in the field, I can say that topology is heavily used in condensed matter physics. It has immense applications!

[@oliviapg]

Started watching this and was like "wait, hasn't he made a video about this before" and then it immediately got to where you talk about how you made a video about this before. The original was already excellent (shown by the fact that despite it having been a few years since I'd seen it, I still remembered at least the general idea of the proof), but this new edition is even better, with a bunch of really nice new insights and animations. Incredible work as always.

[@aliciadevlinder]

Me: I wonder what that surface looks like for a circle Video: "Viewers asked about the circle..." Love this. Merry christmas!

[@Dragon_of_Time]

Relatable

[@kirkelicious]

Video: Shows us the surface for a circle Me: Well, I really could have figured this out myself

[@Dragon_of_Time]

@@kirkelicious True, would've been the most goated pause point to think about it

[@creativecarveciteclimb5684]

Me: "But why can't a klein bottle exist in only 3D without its surface intersecting itself? Grant: "I will leave up on screen an argument for, why..." Me: 😂

[@redstocat5455]

Happy christmas

[@arthurgames9610]

I saw this video on my feed and thought it was like 2 or 3 years old, but it is actually from 9 hours ago!! Pls Grant, never stop making videos, you are revolutionizing the art of mathematical education

[@markbutler94]

This might be your best yet. Absolutely phenomenal example of mathematics teaching.

[@littleblu33]

I gasped when you cut our non-orientable surface into two pieces. I immediately thought, wait oh my god, we CAN put those edges together! Love your visualizations. The conclusion about using 4D space to prove the square case felt completely natural. Less an intuition and more a beautiful conclusion hand-delivered into my lap. Thank you.

[@boydstephensmithjr]

This video made me realizes why Univalence (and HoTT) will be important for programming in the future. In particular, when we want to preserve unordered pairs across a program transformation, and not "flatten" them to ordered pairs and lose fidelity along the transform. HoTT is a kind of "topology on types", and Univalence is an axiom that unifies smooth, bidirectional transformations (isomorphisms, mostly) with equivalence.

[@macronencer]

Wonderful explanation, and I especially loved your closing comments. You have to be one of the most eloquent teachers in the world. Thank you for your gifts.

[@chaoticgood7128]

This makes so much more sense. I thought topology was more of a fringe topic that mathematicians just liked to play with. This video helped me understand that topology has very real applications in lots of different areas of math and consequently many of the sciences.

[@AllenKnutson]

Math prof here (not a topologist). It is in no way fringe. All over math, one creates topological spaces, anytime one has a set of mathematical objects where one can imagine continuous variation among the objects. There's usually lots more structure -- e.g. lengths, angles, volumes -- but if you can't understand even the topology of your space how are you going to face up to its richer structures? First things first: understand its topology, e.g. compute its Euler characteristic, its Betti numbers, its cohomology ring.

[@FelixGigler]

I did my fair share of university courses on topology related stuff, but never encountered such nice motivations for topology. Once again I am in awe of this channel

[@b_dawg_17]

The ending made me think of what I think is called “Category Theory.” The computer programmer equivalent that makes the most sense to me is how the same algorithm can be used to solve different problems. If the root abstract part of the problem is the same, all you have to do is map from one problem domain to the other, then apply the algorithm, then map the answer back to the original domain. It’s this ‘’mapping” step that takes us from questions about inscribed rectangles to Klein bottle or from the traveling salesman problem to any other NP Hard problem. Idk if that made any sense or not 😅 anyways, loved the video! You’ve been instrumental to my mathematical intuitions since highschool, and I can’t wait to see your next video!

[@cainen6355]

What an insanely well made video on this topic. This is a masterclass in not only thorough explanation but especially the excelent use of 3D/2D to map all the ideas into a visual representation using clear markers and colors to encode important information about what we see!

[@rato5611]

It's not an absurd to think that a really big part of the next generation of scientists will have chosen their jobs because of you

[@FlaviaScollo]

Considering I'm one of those people, I agree, it's not absurd at all

[@ross302ci]

I remember lucking out and finding this channel shortly after Grant's very first video. I was an engineering undergrad and wondering how on earth anybody came up with e^i*pi = -1. At the time I was happy to be in engineering but really didn't believe much in my ability to tangle with the harder math and figured I'd just eke out my degree and find a job. Now, decade(ish) later, I have a PhD thanks in no small part to Grant's amazing encouragement and clear explanations of fundamental concepts. Never imagined that would happen. It has been so cool to see this channel grow, and to grow alongside it in my own pursuits. I love to think about the numerous other people who have been similarly impacted by these videos. "Thank you" doesn't even begin to express our gratitude Grant!

[@elijahjamieson668]

I love that you ask us, the viewer, to question you and try to prove you wrong. I know I do not have the knowledge to do so but it gives a nice reassurance that you truly do know what you're talking about (not that I would question it, you are how I got a 5 on AP calc bc)

[@paulolellis5708]

Wow.. mind blown at the mapping on 5:40… what a solution masterpiece! Bravo…

[@n-steam]

I'm no maths genius, but every step you put forward, I was already into thinking down that line. I think that is a credit to your narration and explanation of the problem.

[@tylerrodriguez9935]

So I’m currently at 5:54 and you’re talking about mapping and I just wanna appreciate how you’re able to take these complex math ideas and make them digestible bc I immediately was able to connect the Klein bottle and its shape to this idea of mapping and how topology is a problem solving tool. I believe it’s because if an object is non-orientable and has one side that guarantees there is a pair of points that coincide with the start/end point of the non-orientable shape and we can take that point in x,y,d and extrapolate two line segments using that coordinate data for the midpoint and distance. Not sure if I’m right, but THIS is why I love your videos. It makes things digestible to people with even passing interest and even if I am wrong, I’m still thinking like a problem solver and engaging with concepts that I might otherwise never have known or understood.

[@u2b83]

Thank you! Topology has been the field I've been ignoring all my life but have always been curious about. Your first example reminds me of the time I wrote a program in HS French class to sum the turn angles of a hand-drawn shape, in order to recognize it based on the summation of angles. Obviously that didn't work beyond the first few shapes lol Summing the turn angles of a hand-drawn shape aligns with one of topology's core ideas: understanding properties that remain consistent under deformation. In this case, the sum of the external angles of a simple, closed polygon (like a triangle or square) always totals 360deg, regardless of how the shape looks-an invariant property! Why It Didn't Work Beyond Simple Shapes The program likely struggled with more complex shapes because: 1. Shapes with Multiple Components: If the shape wasn't a simple closed curve, the concept of "turn angle summation" doesn't hold in the same way. 2. Noise Sensitivity: Hand-drawn shapes introduce wiggles and irregularities, making angle sums more erratic. 3. Topological Complexity: Shapes with holes, branches, or overlaps don't fit neatly into the simple sum-of-angles framework. Fascinating topological concepts tied to this idea: Winding Number: This counts how many times a curve winds around a point. For example, you could analyze a hand-drawn loop to determine its "turning nature." Euler Characteristic: A foundational property of surfaces, computed as V−E+F, where V, E, and F are the vertices, edges, and faces of a shape. This is invariant under continuous deformations. Gauss-Bonnet Theorem: For a surface, the total Gaussian curvature relates to the topology, like the number of holes.

[@foolo1]

I have seen so many STEM related KZbin videos so that at 1:47 I thought you were segueing into an ad for Brilliant :)

[@uranium-h3o]

🤣

[@JulienRoigHerr]

Grant does not take sponsors, because he has a pure soul.

[@memena420]

@@JulienRoigHerr are you applying that any youtuber who does take sponsorships has a nasty dirty soul for some reason?

[@jeremiekuchen9779]

I have never seen such a qualitative video, which allows to really grasp what topology is, this bizarre subject that has always been described to me as "generalization of space". Really well done!

[@Konchok_Dawa]

I majored in math and I've always found it so strange how my introduction to topology had almost nothing to do with geometric deformations even though that's how it's motivated in videos like this. I was taught that a topology is purely an extension of set theory, and it amounts to finding structures among the collection of SUBsets of a given set. Subsets that are IN the collection (aka the "topological space") are "open" sets. So when I see videos like this I'm always thinking: what is the underlying set, and what subsets are we talking about? In this case I'm assuming that the plane curve is the main set, while the pairs-of-pairs are the open sets within the *topology* we're constructing (a constraint on the power set consisting of all subsets of the curve). It's just interesting that we can gain an intuitive understanding of a problem like this without ever using the term "open set" but I think it's important for people with topological interest to know, so I'm offering it here.

[@sebastianwesterlund8777]

No, the curve in question has the subspace topology of R^2, which has as a basis the set of open balls in R^2. The reason we care about open sets is because they define what it means for a map to be continuous. When our maps are to and from subsets of R^n one can define this with the usual epsilon-delta definition, but when you want to do constructions like gluing, or to consider spaces way more wild and unintuitive than in this video (say, spaces that cannot be embedded in R^n for any n), the general definition of a topology turns out to be the most useful.

[@Konchok_Dawa]

@sebastianwesterlund8777 I guess I should know that open balls are the elements of the topology of a curve. I guess what I'm stuck on is, in what sense is the first problem in this video a topological problem? Is this 3-d shape a topological space? If so then what are its open sets?

[@Konchok_Dawa]

It seems like what you're saying we just need to consider the 3d figure in the video as a subspace of R^2 or R^3, but if its a 2d, self-intersecting space embedded in R^3 then it's unclear to me which R^n topology is actually helpful

[@Konchok_Dawa]

Hm rewatching the part about the mapping to the torus kind of answers my question I just like to see things axiomatically lol

[@3blue1brown]

What’s interesting, to me at least, is that the initial question is NOT really a topological question, it’s a geometry question. Whether four point make a square is rooted in the metric. But nevertheless, topological properties of the space of loop pairs is relevant to answering it.

[@ymi_yugy3133]

Absolutely amazing. I like how visual and intuitive this proof was. When I was taking topology in college it just felt like algebra. It felt like we spend most of the time talking about definitions of topological spaces and homotopy groups.

[@erenerdem4657]

Thanks for all of your hardwork on this channel.

[@BadChess56]

This comment taught me that 3b1b has now been thanked.

[@AlwxR]

Apart from the math itself and even though the format is different, one learns and gets so much inspiration for teaching uni math by listening to Grant's explanations He also somehow finds the best examples: easy enough to be shown without so much previous knowledge and hard enough to portray many of the harder themes and variations, while also useful for people of all levels

[@kaiwalpanchal5872]

I have been working on some city planning drawings and the scale is not mentioned on them, finding squares will help me find the scale to measure everything on it since there isn't any other obvious way. this video came at the right time!

[@YungKaioken]

I'm an electrician in the commercial field and on my most recent jobs, haven't had any scales on the blueprints either haha kind of important aspect to forget but thankfully being versed in geometry i've managed to figure for them.

[@abcrtzyn]

The first 3b1b video where my jaw dropped. As soon as you mentioned the reflected Möbius strip, I was already thinking Klein bottle and then quickly thought about the fact that it has to self intersect in 3D space. Thanks for having something new when you remake old videos, it really does make them worth watching again.

[@austinclees9252]

Will we be seeing more ‘2nd edition’ videos? I think there’s a lot of value to redoing old video. I watched the first edition of this video back when I was in grade school. now being in university, I’m making connections from this video to ideas I’ve studied/done research with. It’s really cool! And it would be cool to make these new connections to other videos as well.

[@3blue1brown]

I just may do that. Not for every video, of course, but there are some older topics which I love, but which could really use a refresh.

[@smotala11]

@@3blue1brownFourier series by any chance? Feel like there is so much you can do here

[@shedrackjassen913]

Bro just got replied to by 3b1b

[@j.vonhogen9650]

​@@3blue1brown​- This topic definitely needed a 2nd edition, just to prevent people from wasting their time looking for inscribed squares on smooth curves, unaware of the existence of the 2020 (partial) proof.

[@robynrox]

Thank you for the amazing proof! It blows my mind how you can take such a tricky subject and make it understandable. I paused to read your proof and I understood that, too.

[@ainnetwork3308]

This is crazy; an hour ago I came upon a random reel unfolding complicated knots and one comment mentioned the topology science so I went to KZbin to find out more finding your previous video and after few moments I saw the new video with only twenty minutes ago; what concidence; it's look like it was distiny to find out about topology!! Fascinating filed; and beautiful explanation from you.

[@GuyNamedSean]

At 13:50 I got so excited because it all clicked together for me. I literally shouted "that's why we get a Klein bottle!" Topology and geometry are totally my favourite maths. There's so much in it that just clicks together in such a clear way to me.

[@FloydMaxwell]

A problem perfectly suited to your groundbreaking graphics. No wonder you returned to it.

[@jwj410]

Very illuminating - thank you. I think the type of problem you start off with leads you down a topological path, in that you start with ANY curve in a 2d plane and ask if it has a constraining property, such as does there always exist at least one inscribed rectangle. You don't care what the exact coordinate definition of the curve is, just that there is one. So there's a whole family of curves under investigation, and topology allows you to reframe them as other 'things' that you can more easily pin down their constraining properties.

[@dolichakraborty6020]

New topology video feels like a christmas gift ❤

[@julian246810]

Just wanted to say thank you for all the beautiful proofs and ideas you are showing us. As a serious mathematician I really appreciate how precise you make the arguments while also giving people with much less background good ideas of why certain things are true. Truly amazing work.

[@chiaracoetzee]

The moment the Desired Claim about the Möbius strip came up I was absolutely stunned. I could immediately intuitively see that it must be true and how it implied the desired result. This is incredible, even after a whole course on topology none of its problem solving potential was apparent until now.

[@key_bounce]

I really wish I had a teacher with your graphical approach back in college. I've noticed in school that as long as people are doing things that has graphical grounding, people learn the math. But when you go from the number line, to ... anything else, the pictures are dropped, and just "here's how you manipulate the numbers", with no real reason or understanding. I was fortunate enough to have graphical integration and differentiation in high school (AP program), so I understood what was going on. And my progress in math pretty much stopped when the graphical understanding stopped. Our college topology was all abstract and properties of nearby points, and I don't even remember the names attached to the spaces that had certain properties. You make advanced math topics understandable. Thank you.

[@szilardecsenyi516]

I know an area where this mathematical problem would help to make an exciting discovery. As we know, in quantum physics there is no chaos, a linear differential equation describes each behavior of each particle. This is why the symmetry of the system determines the probability at which points in space it can be located, where 0 is called quantum scars. If it were guaranteed that squares always appear in the phase space on the wandering paths, then it would also justify why high-symmetry microstates can resolve the Loschmidt reversibility paradox in ergodic systems. Interestingly, if the entropy increase and temperature change slow down, this may be even more pronounced, the consequence of which may be that the equilibrium state of the biosphere is stuck at a relatively high temperature and is not "forced" to increase its entropy indefinitely, because the set of available microstates has high symmetry, the phase space wandering passes into the world of a ball bouncing back and forth on a billiard table

[@hoebare]

If someone had asked me, before reading this comment, "do you understand English", I would have said "yes".

[@FunIsGoingOn]

So that equation describes a theoretical situation where everything is in the phase space as expected, but in practice it isn't? Well then welcome to the universe, so there actually is chaos even in quantum physics.

[@daviddickson3600]

This was such a fulfilling video to follow along. Guiding through a seemingly too-difficult (in my eyes) problem intuitively with topology like mobius strip, torus, and klein bottle. The reveal of each topology had me so excited. Thank you 3b1b for making me see topology in a new way

[@makiaveluz]

25:35 this reminds me of the fractal video. How it was thought as a tool to encode roughness but became a kind of mathematical toy.

[@Khashayarissi-ob4yj]

So beautiful. Hoping for more videos. To step onto the path of knowledge and this is what a beautiful path. To share the joy of knowing with others and this is what a beautiful feeling. May God bless your lives.

[@manstuckinabox3679]

According to the comment section, 3b1b is not my youtube math teacher, he is the youtube math teacher of my youtube math teachers.

[@abramosia]

As a math teacher, I concur.

[@izzywakefield72]

I am a design engineer who works with complex surfaces just like this every day. Thank you for explaining a bit of the math behind how my software works!

[@eclipticsim]

Andrew Lobb gave linear algebra lectures to me in first year

[@fredgoodyer4907]

He was my geometric topology tutor and algebraic topology lecturer 😊 A legend and a meme for my year group, he became the pp of the maths gc 😆

[@danieledgar442]

Lobbster mentioned

[@wolkensaft]

Your videos always make deep mathematical insights not only approachable, but understandable. Thank you for that!

[@TheReligiousAtheists]

Regarding the embedding of the Mobius strip with circular boundary, one can realise that such an embedding exists using a more abstract approach (in particular, without directly constructing the embedding). There's a theorem in knot theory that if two knots (closed loops of string) can be deformed into one-another, then you can also deform all of space so that one knot lands on the other. The boundary of the 'standard' embedding of the Mobius band is a string with a twist in it, and undoing that twist gives a circle. Hence, by the theorem, you can now deform all of space (including the Mobius strip sitting inside it) so that the boundary becomes a circle. Of course, the proof of the theorem is not easy and takes a lot of care. However, this theorem is used basically all the time in knot theory, so for someone familiar with the area it gives an easy way to see why the claim at 18:02 is false.

[@silvera1109]

Wow. Finally some intuition for mobius strips and tori! I love how you make topics simpler with fabulous visualisations. Thank you so much for your content Grant. Love this.

[@StefanoBorini]

You know what I realised? that you never really explained how a mathematicians puts down the findings. I mean, imagine that you were to do this kind of discovery on your own. Would you write it down as you think about it? would you invent some notation, or use some standard notation, during your thought process. How, in other words, do you move your thought process into paper, when you do research math?

[@hectormartinpenapollastri8431]

Mathematician developed precise notation to write all these unambiguously. It is literally a language in itself

[@Ryan_Thompson]

It depends on the branch of math. Every bit of math I've done professionally (or recreationally) that was non-trivial, ended up being a journey through different types of notation (both standard and *very* ad-hoc... literal napkin doodles, sometimes!) Or writing a short (or long) computer program to get an intuition for something, if not an exhaustive solve. Write *something* down when you can, but if there's one thing I've learned, it's to *not* get hung up on notation during the discovery phase--just do whatever makes sense. And if it's easier to start gluing construction paper together, do that instead, and take a picture or something. Worry about precise notation when you publish. Until then, use as much or as little as you need to work efficiently.

[@sebastianwesterlund8777]

This is a very good question, and indeed some of the most important math discoveries of the 1900s (or indeed in the history of math!) have been not about new proofs of theorems or solutions of problems, but rather about inventing new definitions that allow researchers to talk more clearly about the things they already "felt" were true but had a hard time putting into words. I mean, just think about what it must have been like when algebra was first invented. Surely people would have been solving what we'd consider as "algebra problems" for hundreds if not thousands of years at that point, but the ability to put it down in symbols just allows you to reason in a much more clear (and therefore powerful) way.

[@MK-13337]

Well notation in the moment of research is usually just letters and symbols. Some letters and symbols are things people hava defined before and those have know notation but some are just your own and you call them whatever is convenient. "We have this subset of R^n, let's call it A and we want this other set, let's call that B to be a set with smaller area but same volume" or something like that. And then you just draw 2 "random" blobs on paper called A and B and you start to think what would a shape need to have the least surface area with a set volume (and you quickly end up with a sphere).

[@headlibrarian1996]

@@MK-13337 I assume we have a proof for this general proposition, that in N dimensions the object having lowest surface area is always the N-sphere? Or is this an open problem like the Poincaré Conjecture was?

[@BecomingHypnos]

I was hoping for an answer but I ended up having more questions. As someone who studied civil engineering and doesn’t know much about math, this kept me severely intrigued and I loved the way you revealed each of the three shapes (It was like watching a movie). Have always loved your videos, don’t stop making them

[@TheSummoner]

1:34 read: "as well as humanly possible"

[@RemedyRodriguez]

This video, like many other of the videos on your channel has fueled my curiosity and interest math. You explain topics in a way that anybody can understand which i highly appreciate as a young person. thank you so much.

[@Vengemann]

idk why but Topology always reminds me of THAT one video where they say "NO creases and NO tears to get surface outside in" it'll ruin the game lol

[@matthewgiallourakis7645]

@@Vengemann Inverting the sphere?

[@Vengemann]

@@matthewgiallourakis7645 yeah

[@chaken6187]

@@Vengemannoooo that one!

[@cfaibah]

Hey I remember that video. I think that one got a bunch of people interested in topology - even if superficially - because the problem is simple enough to understand but not really intuitive to solve - and it has these beautiful early cgi animations

[@jaco621]

my mind was totally blown when the mobius strip was formed. the torus was mind blowing enough, but the mobius strip forming was just *chef's kiss*

[@yahav897]

I'd LOVE to see a video on number theory

[@thereal_pixel_phantom]

Yes please

[@BadChess56]

Number theory is beautiful.

[@bobopokomono-nu3gv]

indeed, about time he leaves the riff-raff behind...

[@MehdiSahtali-k8g]

Some topics in the Geometry of Numbers (geometric interprztation of number theory developed by Minkowski) would probably fit really well with Grant's preference for surprising solutions bridging seemingly very different parts of math. And much of it can probably be animated.

[@KupoPallo]

I took topology 1 and 2 before rage quitting that branch, yet I learned more from this shorter than 30 min video. Merry Christmas everyone and thanks 3B1B for being amazing.

[@GlobalScienceNetwork]

I like the Mobius strip concept and how it creates continuous access to the top and bottom of a plane. The rest I did not really get. It seems like you can create most 2D shapes from a continuous curve. If these shapes have a physical meaning then calculating them provides insight.

[@philipwilkie3239]

The animations literally bring tears to my eyes! I can't explain it, but somehow to me this is proof of the existence of a reality beyond the mundane.

[@Wepaaaaaa]

My bed sheets at 3 am

[@chnyuk]

I love how the given problem that was mentioned as one of your favorite math problems, turned out to be one of infinitely many applications of topology as a tool, which (topology) in turn is a game of finding associations between things, as the video stated. math is beautiful

[@BurkinaFasoful]

Man! That sheds some light on a 25 year old "bar" question :) (if I can call it that) When we were math students back than, we were (of course :) ) drinking in many different bars and places, a lot of them with uneven floors. So trying to fix our chairs in a way they don't rock, especially those tall bar stools (or whatever they are called, excuse my english, it's not native for me), that question arised - is there a floor so uneven that is not possible to put a 4 legged chair on it without rocking? So if you cut the floor with a plane, you'll get one (or more) of these closed continuous curves. Ok, the points where the legs of a chair touch the ground are rather "square" and definitely not an arbitrary rectangle and you can put a leg on other, not connected part of the cross section , but that's an important step and gets us closer :) You say that you don't know about any practical application, but believe me trying to balance your bar stool especially in 3 a.m. ... that can become even a life saving knowledge :) (but seriously, I guess there's good number of cases in which a complex surface have to be supported/connected on 4 supports of equal length or something) That's something that I love about math - even drinking in a poorly lit basement with an uneven floor can lead to questions that involve some pretty interesting math constructions and theories :) btw. Love your videos, thanks for doing them.

[@vinven7]

what a great example!

[@grACE-tm9ft]

watching your videos, compared to other forms of entertainment, is a high hanging but sweeter fruit for me. Big thanks!

[@BadChess56]

This video taught me that an open problem taught 3B1B what topology is.

[@G.Aaron.Fisher]

Glad you did a version 2. I liked the first video, but this beautifully addresses a couple of the unsatisfying elements of it.

[@JTCF]

If you take a slice of uneven terrain, the border between terrain and air would be such a curve. And a rectangle/square could represent the four legs of a table. This kjnda connects to being able to put a table on a surface and it won't wobble around and will be parallel to the ground (gravitationally). Also one simple trick to get a 4-leg rectangle to not wobble on uneven terrain is to rotate it. At some point it will stop. I use it for my weight scale because my floor is made of badly warped wood.

[@Sleeping_Aizawa]

Your My Hero.. I was confused where to start and now I understand I never understood the abstraction until now.. Thank you so much.. The Holography was amazing. I got so many new ideas on how to approach problems and now I am getting even more.. I love the math and visualization for those who are unable to understand the abstractness without a representation on why and how its important..

[@isothepiano]

The animations are astonishing

[@eriktempelman2097]

Indeed. Makes one think how topology was done BEFORE the advent of such animations.

[@kirkelicious]

What's even more mindblowing, is the fact that he himself coded the animation tool from scratch in python.

[@maxd1744]

Draw, visualise, go crazy (or try to not go crazy). ​@eriktempelman2097

[@isothepiano]

​@@eriktempelman2097Yeah, exactly

[@isothepiano]

​@@kirkeliciousikr, I am a programmer myself and I can definitely tell coding a reliable python library is not an easy task, let alone coding a graphics library

[@Deeco_Inc]

Your passion for math and geometry is contagious and inspiring. Keep up the great work!

[@Subsessor]

4:15 i dont get it. why do both endpoints have to move to find all possible combinations of "two endpoints"? wouldnt it be a simpler example to let just one endpoint move along the curve and only move the other one until we finished the whole curve?

[@mjjjjjjjjjjjl]

I think it's just for the sake of animation, it makes for prettier visuals. You're also describing moving both points, but just moving the second point less often.

[@christopherlocke]

You cannot move a point along a continuous line just a little bit. There will always be infinite points in between. If you have numbers like 1, 2, 3, 4, you could take pairs like 1,1 then 1,2 then 1,3 then 1,4 then shift the first to get 2,2 and so on. But this only works because these numbers are discrete and finite.

[@alstonsmith3924]

Seeing an animated video likes this makes me appreciate the sheer genius of those who developed this theory without these kinds of visualizations

[@chessematics]

25:57 Bach is smiling with his crab canon right now.

[@nerdiconium1365]

This also reminds me of a video from way back that George Hart did for the Simons Foundation about representing 2 note chords with a möbius strip.

[@dodaexploda]

If anyone has the money and hasn't done it yet, I highly recommend buying an Acme Klein bottle from Cliff Stoll. The process is lovely and fun. He takes photos with his bottle in the garden before sending them to you. Adds hand drawn art work to the box, the correspondence is lovely. Just everythign about it is amazing. Like I'm in Canada so he made sure all of his creative writings on the box were also in French. The email correspondence included info about the weather. All stuff that takes a good amount of work. I just highly highly recommend it. The bottle is neat, but the experience is not something you're going to have forever.