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 :)
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
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?
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?
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.
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.
@@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)
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
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.)
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.
@@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.
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.
The way the videos on this channel activate my neurons is unlike any other channel on TH-cam. Each new video only further cements this channel being the best educational channel on TH-cam.
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.
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.
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.
@@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.
@@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.
@@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..
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.
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.
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.
@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?
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
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.
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.
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 👏
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."
i watched the original video finishing highschool, i just did an undergrad in math, and know i see the second edition. i feel trick about the fact that the problem being unsolved. if it was solved for smooth curves, that's good enough for me, it is already solved! (half joking). i like avoiding the definition of a topological space. it is important to work with that formalism, of course, but it is an awful introduction. still, i remember on your video of limits you said something like "knowing the idea of approaching intuitively is enough for intuition somehow, but it is important and nice to get to know the definition". i think it would be nice to enphazise that, and that working with topology is closer to the formal definition, than these (very great and informative) animations. i really like that you said that a topological space isn't really the space itself, but the class of all spaces homeomorphic to it. it is technically false, but great for teaching. i remember my differential geometry professor saying in a thick russian accent that "defining two manifolds as the same set, with the same topology anf the same atlas, is a very primitive definition". it is nice to remark that, and i like how you explained it at the end. i loved the original video for how it introduced me to these ideas, and know knowing them, i really like it as an introduction for the new audience. i love seeing this edition as a someone who would teach this, years after seeing the firtst edition while learning.
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 TH-cam 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.
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: 😂
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!
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
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.
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.
@@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.
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
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.
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- 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.
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.
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.
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
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?
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.
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.
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).
@@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?
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!
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.
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
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)
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.
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!
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.
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...
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.
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!!
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.
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.
Just to get this out of the way: I feel like the question of "what purpose does this serve" is something that will often be answered in the future. Maths has this weird way of becoming useful in one way or another. Maybe some step or method will be useful to be applied in another problem or maybe another problem can at some point be partially reduced to this one. And sometimes, like magic, multiple of these "useless" areas of maths will come together and suddenly we have a hypothesis about a function that could predict primes. Sure, this is an easy example but something that never ceases to amaze me about maths is how in effectively everything there ends up being some tiny aspect that is weirdly "useful" either in other areas of maths or in related fields
I discovered your channel today and I love your work. Your neural net video on recognizing digits has inspired a new assignment for my high school MATLAB students.
I have been a follower of your channel for many years. This episode is one of your best. It brings together so many topologic ideas into solving a seemingly simple question. I for one just loved every minute of this video. Also the graphics are in a world of there own. Thanks, so much for the gifts you bestow on us.
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!
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.
Topology is essentially the study of properties of space that are preserved under continuous transformations, such as stretching or bending, but not tearing or gluing. It's a fascinating subject because it deals with abstract concepts like continuity, limits, and boundaries.
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.
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.
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!
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..
Witten's knot theory of 4D dynamic space time is 3D quantum black hole at Planck scale l=g*m/c^2=1.616*10^-35 meter, proton scale pl=g(p)*(4pi*pm/3)/c^2=8.809*10^-16 meter, Atom scale A^2=g(p)*pi*me/128.4980143*c^2 by super symmetry 137.036=g*m^2/k*e^2=GR/QM=(m*g*pm*g(p)*pm*g*m)*(1/137.036)/((e-)*g*(e+))=ER/EPR oscillating between those 3 3D QBH form 4D Mobius bottle, by ADS/CFT those 3 3D QBH can turn into 2D Mobius stripe ch=2pi*l*m*c^2=2pi*pl*pm*c^2/4.188=2pi*A*me*c^2/137.036 connect by strong force g(p)=g*m^2/pm^2=g*(pl/4.1888*l)^2=1.13*10^28, EM force between proton[pm=1.672621868*10^-27 kg], electron[me=9.10938356*10^-31 kg] in hydrogen Atom[A=5.29177282*10^-11 meter] k*e^2=g(p)*pm^2/137.036=ch/(2pi*137.036)[e+]=me*(c/137.036)^2*A[e-]=4pi*g(p)*pm*me*137.036/128.51991, weak force pm/me=1836.1527=4pi*137.036^2/128.51991=1/(4*A1*137.036^3)=(pi/(128.4980143*A*137.036))^0.5, which can turn into 1D string ((8.753/8.45)-1)/(A/A1)-1)=2*105 between QCD of muon[105 mev=(3*137.036/2)*0.511], QED of electron[0.511 mev=me*c^2=g*m^2*137.036/A] for Yang-Mills gauge field with same ratio for 4D dynamic space time of 3D Mobius bottle, from it can deduce (me/pm)^2/137.036=128.4980143*A/3.1415926=2.16*10^-9=0.00116592026-0.00116591810 :discrepancy of muon magnetic moment of (g-2)/2 factor between experiment data[0.00116592026+((61-41)+(57-25))*10^-9/2), theoretical prediction[0.00116584719+6845*10^-7+154*10^-8+92*10^-9] from Fermilab at 8/10/23 only can get this result fron 1D string, 85.73% ratio between muon, electrong decay from strong force by 100-85.73=14.27=A/(3*137.036*(A-A1))=128.51991/(3*137.036*(128.51991-128.4980143))=128.4980143/3^2 deduce A1/A=128.4980143/128.51991 : oscillation of Atom radius cause Einstein's Brownian motion of 1D string we can observe 4D dynamic space time of Witten's knot theory in action, deduce 2^(1/6)*ch=125.0895 Gev : Higgs boson from extra 6 dimensional Calabi-Yau manifold 2pi*k*e^2=2pi*g*(137.036*e-/l)*(137.036*up*e+/l)=ch=8pi*g*(m*c^2/2)^2/c^4=2pi*l*m*c^2[type1]=2pi*pl*pm*c^2/4.1888[type2b]=2pi*A*me*c^2/137.036[type2a] by super symmetry 1/137.036=e^2/(2*ch*p)=k*e^2/g*m^2 unite QM with GR.(c=1/up)^0.5=299792458[meter/second], u=4pi*10^-7, p=8.85418782*10^-12, m=(ch/2pi*g)^0.5=2.176466*10^-8 kg].
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
So beautiful. When you brought out Chekov's gun after mentioning torus, Möbius strip, and Klein bottle, I know where you're going but still amazed with your way of unpacking it anyway.
Your original video on the inscribed rectangle problem was the first 3B1B video I ever watched and I was immediately hooked :) it's hard to believe it's been almost a decade
My favorite example of topology are game maps. The game Civilization is set on a Cylinder, because you can't cross the northern and southern edges of the map but you can cross the eastern and western edges of the map, and when you cross the edge you end up at the same latitude on the other side. If you crossed and ended up at the opposite lattitude, you'd be on a Mobius Strip. You could hypothetically mod civ to be any topological surface.
This was truly an amazing video. I agree with many comments repeating the title: "This open problem taught me what topology is". Huge thanks for this! I've always felt I do not really know what topology is about. Perfect christmas present with the shapes! :)
I had always struggled with understanding this until I heard the question(s) below. Give them a go after watching the video, in this specific order, you'll love them. 1- A monk leaves to ascend to the temple on top of a mountain at 9am and arrives at 5pm. The next day he leaves the temple at 9am and arrives back at the foot of the mountain at 5pm. Is there a point in time where he was at the same location on the path at the same time? 2 - Prove that on a 2D earth, there exists a diameter such that the temperature at the endpoints is equal. 3 - Prove that on a 3D earth, there exists a diameter such that the temperature and humidity at the endpoints are equal. Finally... 4 - Does every convex closed curve in the plane contain all four vertices of some square?
Moreover in a new paper just posted a few weeks ago they show that every cyclic quadrilateral can also be found in every smooth curve! great video as always
Merry Christmas Mr Grant! Thanks not only for the great content, but for your support of the maths and STEM community here on youtube, specially summer of Math.
Oh, man. You are so good at providing intuitive ways to think about complex topics! The feeling I got when I predicted you were eventually going to map things to a Klien bottle surface when you first mentioned the Chekov's gun bit was super cool!
Okay - be kind with me but there are some things , ideas really, I would like to bring to comments section about this video. They are meant positively and hopefully they will be taken positively. 1 - first there are not really 4 dimensions: x, y, d and theta. There is a mashup of 4 variables in 3 dimensions. Orthogonality took a dive somewhere. Alternatively look for growth in two directions simultaneously (see 3 below) 2 - there is no intersection problem in Klein's bottle. Taking a linear progression along [0, 1] we can put the "intersection" anywhere in that range. Then the bottle can grow from anywhere in from zero to 1 and/or from 1 to zero. Preferably a distance away from the endpoints BUT the bottle grows in both directions simultaneously Seeing it that way there are no problems at x=0 and/or x=1 otherwise at the endpoint x would simultaneously be zero and one. This allows x to grow increasingly from a closed zero in direction of 1 AND x to grow decreasingly from closed 1 in direction of closed zero. The growths do not intersect they merely meet. I suppose it could be argued that there has to be a join somewhere but as that would be infintessimally close at infinitely many points on a smooth surface who would notice the join? 3 - linear growth in one direction is fine but linear growth in two directions is even better 🙂
I really like the way you keep me interested and in love with math even when life is trying to distract me from its beauty Thank you 3B1B and Merry Christmas!
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.
I always thought that topology was the weirdest, somewhat useless (but at the very least still kinda neat) branch of mathematics until I started taking analysis classes. Suddenly topology seems like one of the single most important and fundamental branches of math.
Oh. Oh this is wonderful, I can't like immediately think of applications but the notion of mapping constraints on some set of values to a topological class that we can say things about dose seem useful. Coming from engineering I think a lot about how we represent physical constraints like the shape and loading of materials or the limit of inputs to a control system. Additional the, restricting a set of continuous points to a unit square and then showing things about sets of points in that space reminds me a lot of statistics and spesificslly statistical mechanics where things like particle location and interaction can be represented similarly. This, this is good.
Topology has interesting and productive core intuitions, but as long as we try to speak of topology in terms of "points" and "neighbourhood", "point" in that language remains "undefined primitive notion" and as such antimathematical dishonesty, just like Weierstrass' "nearness" (which is regularly presented in scary quotes for the obvious reason that it is incoherent antimathematical blather), we can't solves problems caused by mathematical dishonesty by remaining foundationally dishonest and pretending that the Zeno-absurdity of point-reductionism can make any genuine mathematical sense. Hint: to solve the rectangle problem, study origami. The most important topological evolutionary step after straight edge and compass.
18:50 wouldn’t flattening the mobius strip edge to the 2d circle just show the example of the infinitly close intersection for circle (i mean the dome in 7:47)?
I remember being presented with this argument when I was in 10th grade over 10 years ago and not liking it very much, because the fact of the non-embedding of the Klein bottle to me is as non-intuitive as the desired rectangle result itself. Speaking of which, the argument at 22:30 works only in the smooth category (you implicitly use tubular neighborhood etc), and although the fact itself is true for continuous embeddings, I don't think there's an elementary argument there to exclude wild embeddings (and quick googling leads me to the mathoverflow question which basically agrees with this assessment).
The mobius strip surface is the coordinate system of all possible non-unique point combinations. Each point on the strip represents a pair of points on the 2d shape. I thought the next step would be to plot "d" (the distance between those two points) as some "height" that is "above" the mobius strip. It seems like *that* surface (the "inflated mobius") is what the "weird architecture" surface maps to, and any spot on the "inflated mobius" where the heights are the same would represent two (x,y) pairs that generate an inscribed rectangle.
Amazing video.. i do however want to point out that you elude to the idea of homeomorphisms and suggest they are the continuous maps but as im sure you know they do have 2 additional constraints.. it is a creative decition on your part for sure but i wouldve loved to be mentioned that its not just continuity that gives this equivalence
One interesting idea could be to find a way to relate the rectangles in a circle to the ones on a given shape, then possibly you could find similar topological consistencies with the fractal shapes, like how does deforming a circle and its rectangles/square map to the snowflake And with fractals to, if you can prove in the case of the snowflake that the first iteration has rectangles/squares, then the subsequent ones will most likely do to based on what parts are fractaling, likewise there probably a proof idea for infinite squares in self similar fractals, like the snowflake (i could see something like the Julius set possibly being harder, as it changes as you zoom in, where as the snowflake doesn’t)
"I'll talk about smoothness in a moment" Oh I think spline continuity and smoothness has been embedded deep into my brain since the adventures I had tryibg to make a perfect procedural 3d train/rollercoaster track
This reminds me a lot of my PhD studying the mid-point Locus, symmetry sets, the rotational symmetry set. For the mid-point locus you take pairs of points on a curve, such that they both lie on a circle which is tangent to the curve at both points. In the mid-point locus for each such pair of points you plot the mid point. For the symmetry set, you take the center of the circle. In both cases the pairs of points have a local reflectional symmetry. The rotational symmetry set, which I studies captures local rotational symmetry. In all three cases the set of mid-points/centers creates a new curve, that in some sense capture the overall shape of the curve. While there may not be an application for inscribe squares there is a lot of interest in mid-point locus and symmetry set in the field of computer graphics. My main task was writing code to calculate and draw these curve. Here I pretty much had to figure out the topology, a Möbius strip, like you have done, and then ensure algorithms worked across the joins.
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 :)
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
for one it highlights the necessity of rigorous proves. Things that make sense "if you think about it" aren't always true.
Analysts just need one dimension. Enjoy your continuous everywhere, differentiable nowhere function
@@gabrielvieira3026 AAH YES THE TROLL MASTERGUIDE
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?
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
Yeah...
Merry christmas
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?
You both should do a lecture together. 1.5 hours. On a strangely interesting topic.
Obviously. Algebra is fake math.
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.
The exact thing I wanted to comment, 2 minutes in
@@benj6964This comment articulates well what I have in mind when watching 3b1b.
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.
@@johnsmith1474 no, not the pretty moving graphics, art as in the art of teaching
@@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)
New editions for old videos is a wonderful wonderful idea!
I KNEW my deja vu was founded!
I also remember hearing that this problem was solved
Happy christmas
Agreed!
YES! The animations are top-notch!!
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
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.)
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.
@@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.
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.
Happy christmas
The way the videos on this channel activate my neurons is unlike any other channel on TH-cam. Each new video only further cements this channel being the best educational channel on TH-cam.
I loved my Christmas gift. Everybody would love having a topology class for Christmas
Merry Christmas evening!
@@S.G.W.Verbeek But now is 3 pm
@@marcoazevedo78857 pm here😅😆
It's 1:16 pm christmas eve for me...
Some of us want family and money
This video has now taught me what topology is.
This comment has now taught me what topology is.
I still don't know
@@BatSignalJammer This reply has now taught me what topology is
@omlett6482 this replay has now taught me what topology is.
This video has now taught me that i was wrong about knowing what topology is.
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.
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.
Mine was the huggbees version. Equally interesting for different reasons
My first instinct when I saw the mobius strip to circle mapping was “but he pulled a loop tight and that’s cheating!”
Happy christmas
@@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
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.
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.
@@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.
@@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.
@@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..
This simple looking piece of math also has connections to why you don't fall through the floor.
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.
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.
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.
@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?
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
Hm rewatching the part about the mapping to the torus kind of answers my question I just like to see things axiomatically lol
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.
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.
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 👏
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
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."
i watched the original video finishing highschool, i just did an undergrad in math, and know i see the second edition.
i feel trick about the fact that the problem being unsolved. if it was solved for smooth curves, that's good enough for me, it is already solved! (half joking).
i like avoiding the definition of a topological space. it is important to work with that formalism, of course, but it is an awful introduction. still, i remember on your video of limits you said something like "knowing the idea of approaching intuitively is enough for intuition somehow, but it is important and nice to get to know the definition". i think it would be nice to enphazise that, and that working with topology is closer to the formal definition, than these (very great and informative) animations.
i really like that you said that a topological space isn't really the space itself, but the class of all spaces homeomorphic to it. it is technically false, but great for teaching. i remember my differential geometry professor saying in a thick russian accent that "defining two manifolds as the same set, with the same topology anf the same atlas, is a very primitive definition". it is nice to remark that, and i like how you explained it at the end.
i loved the original video for how it introduced me to these ideas, and know knowing them, i really like it as an introduction for the new audience. i love seeing this edition as a someone who would teach this, years after seeing the firtst edition while learning.
I have seen so many STEM related TH-cam videos so that at 1:47 I thought you were segueing into an ad for Brilliant :)
🤣
Grant does not take sponsors, because he has a pure soul.
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 TH-cam 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.
Me: I wonder what that surface looks like for a circle
Video: "Viewers asked about the circle..."
Love this. Merry christmas!
Relatable
Video: Shows us the surface for a circle
Me: Well, I really could have figured this out myself
@@kirkelicious True, would've been the most goated pause point to think about it
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: 😂
Happy christmas
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!
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
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.
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.
@@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.
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
If someone had asked me, before reading this comment, "do you understand English", I would have said "yes".
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.
Can we just take a moment to appreciate the amazing visuals here? This video had some gorgeous animations!
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.
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.
@@3blue1brownFourier series by any chance? Feel like there is so much you can do here
Bro just got replied to by 3b1b
@@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.
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.
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.
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
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?
Mathematician developed precise notation to write all these unambiguously. It is literally a language in itself
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.
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.
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).
@@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?
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!
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.
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
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)
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.
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!
Andrew Lobb gave linear algebra lectures to me in first year
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 😆
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.
A problem perfectly suited to your groundbreaking graphics. No wonder you returned to it.
Wow.. mind blown at the mapping on 5:40… what a solution masterpiece! Bravo…
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...
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.
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!!
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.
Thanks for all of your hardwork on this channel.
This comment taught me that 3b1b has now been thanked.
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.
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.
I love how this is also a Christmas special edition : D
Just to get this out of the way: I feel like the question of "what purpose does this serve" is something that will often be answered in the future. Maths has this weird way of becoming useful in one way or another. Maybe some step or method will be useful to be applied in another problem or maybe another problem can at some point be partially reduced to this one. And sometimes, like magic, multiple of these "useless" areas of maths will come together and suddenly we have a hypothesis about a function that could predict primes. Sure, this is an easy example but something that never ceases to amaze me about maths is how in effectively everything there ends up being some tiny aspect that is weirdly "useful" either in other areas of maths or in related fields
This is a great Christmas gift to humanity
I discovered your channel today and I love your work.
Your neural net video on recognizing digits has inspired a new assignment for my high school MATLAB students.
I have been a follower of your channel for many years. This episode is one of your best. It brings together so many topologic ideas into solving a seemingly simple question. I for one just loved every minute of this video. Also the graphics are in a world of there own. Thanks, so much for the gifts you bestow on us.
New topology video
feels like a christmas gift ❤
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!
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.
Words can't express the level of brilliance- of the subject, the manner in which it was presented here and all the ideas along the way!
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.
Topology is essentially the study of properties of space that are preserved under continuous transformations, such as stretching or bending, but not tearing or gluing. It's a fascinating subject because it deals with abstract concepts like continuity, limits, and boundaries.
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.
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.
Oh wow, thanks! Do you have any favorite golden keys that came up in this “higher” courses?
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!
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..
Witten's knot theory of 4D dynamic space time is 3D quantum black hole at Planck scale l=g*m/c^2=1.616*10^-35 meter, proton scale pl=g(p)*(4pi*pm/3)/c^2=8.809*10^-16 meter, Atom scale A^2=g(p)*pi*me/128.4980143*c^2 by super symmetry 137.036=g*m^2/k*e^2=GR/QM=(m*g*pm*g(p)*pm*g*m)*(1/137.036)/((e-)*g*(e+))=ER/EPR oscillating between those 3 3D QBH form 4D Mobius bottle, by ADS/CFT those 3 3D QBH can turn into 2D Mobius stripe ch=2pi*l*m*c^2=2pi*pl*pm*c^2/4.188=2pi*A*me*c^2/137.036 connect by strong force g(p)=g*m^2/pm^2=g*(pl/4.1888*l)^2=1.13*10^28, EM force between proton[pm=1.672621868*10^-27 kg], electron[me=9.10938356*10^-31 kg] in hydrogen Atom[A=5.29177282*10^-11 meter] k*e^2=g(p)*pm^2/137.036=ch/(2pi*137.036)[e+]=me*(c/137.036)^2*A[e-]=4pi*g(p)*pm*me*137.036/128.51991, weak force pm/me=1836.1527=4pi*137.036^2/128.51991=1/(4*A1*137.036^3)=(pi/(128.4980143*A*137.036))^0.5, which can turn into 1D string ((8.753/8.45)-1)/(A/A1)-1)=2*105 between QCD of muon[105 mev=(3*137.036/2)*0.511], QED of electron[0.511 mev=me*c^2=g*m^2*137.036/A] for Yang-Mills gauge field with same ratio for 4D dynamic space time of 3D Mobius bottle, from it can deduce (me/pm)^2/137.036=128.4980143*A/3.1415926=2.16*10^-9=0.00116592026-0.00116591810 :discrepancy of muon magnetic moment of (g-2)/2 factor between experiment data[0.00116592026+((61-41)+(57-25))*10^-9/2), theoretical prediction[0.00116584719+6845*10^-7+154*10^-8+92*10^-9] from Fermilab at 8/10/23 only can get this result fron 1D string, 85.73% ratio between muon, electrong decay from strong force by 100-85.73=14.27=A/(3*137.036*(A-A1))=128.51991/(3*137.036*(128.51991-128.4980143))=128.4980143/3^2 deduce A1/A=128.4980143/128.51991 : oscillation of Atom radius cause Einstein's Brownian motion of 1D string we can observe 4D dynamic space time of Witten's knot theory in action, deduce 2^(1/6)*ch=125.0895 Gev : Higgs boson from extra 6 dimensional Calabi-Yau manifold 2pi*k*e^2=2pi*g*(137.036*e-/l)*(137.036*up*e+/l)=ch=8pi*g*(m*c^2/2)^2/c^4=2pi*l*m*c^2[type1]=2pi*pl*pm*c^2/4.1888[type2b]=2pi*A*me*c^2/137.036[type2a] by super symmetry 1/137.036=e^2/(2*ch*p)=k*e^2/g*m^2 unite QM with GR.(c=1/up)^0.5=299792458[meter/second], u=4pi*10^-7, p=8.85418782*10^-12, m=(ch/2pi*g)^0.5=2.176466*10^-8 kg].
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
@@Vengemann Inverting the sphere?
@@matthewgiallourakis7645 yeah
@@Vengemannoooo that one!
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
So beautiful. When you brought out Chekov's gun after mentioning torus, Möbius strip, and Klein bottle, I know where you're going but still amazed with your way of unpacking it anyway.
According to the comment section, 3b1b is not my youtube math teacher, he is the youtube math teacher of my youtube math teachers.
As a math teacher, I concur.
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*
Your original video on the inscribed rectangle problem was the first 3B1B video I ever watched and I was immediately hooked :) it's hard to believe it's been almost a decade
My favorite example of topology are game maps. The game Civilization is set on a Cylinder, because you can't cross the northern and southern edges of the map but you can cross the eastern and western edges of the map, and when you cross the edge you end up at the same latitude on the other side. If you crossed and ended up at the opposite lattitude, you'd be on a Mobius Strip. You could hypothetically mod civ to be any topological surface.
Another masterpiece, what a nice Christmas gift! Thank you !!!
We need a 3b1b series on topology!
This was truly an amazing video. I agree with many comments repeating the title: "This open problem taught me what topology is". Huge thanks for this! I've always felt I do not really know what topology is about. Perfect christmas present with the shapes! :)
Seeing an animated video likes this makes me appreciate the sheer genius of those who developed this theory without these kinds of visualizations
23:45 question: how did you numerically find all of these examples for this specific curve?
Probably ran some code to do it
I had always struggled with understanding this until I heard the question(s) below. Give them a go after watching the video, in this specific order, you'll love them.
1- A monk leaves to ascend to the temple on top of a mountain at 9am and arrives at 5pm. The next day he leaves the temple at 9am and arrives back at the foot of the mountain at 5pm. Is there a point in time where he was at the same location on the path at the same time?
2 - Prove that on a 2D earth, there exists a diameter such that the temperature at the endpoints is equal.
3 - Prove that on a 3D earth, there exists a diameter such that the temperature and humidity at the endpoints are equal.
Finally...
4 - Does every convex closed curve in the plane contain all four vertices of some square?
Moreover in a new paper just posted a few weeks ago they show that every cyclic quadrilateral can also be found in every smooth curve! great video as always
This just might be the most beautiful display of mathematical concepts I have seen on this channel to date
Merry Christmas Mr Grant! Thanks not only for the great content, but for your support of the maths and STEM community here on youtube, specially summer of Math.
This video was better than any other Christmas gift I got today. Have an amazing Christmas and a happy new year everyone!
Oh, man. You are so good at providing intuitive ways to think about complex topics! The feeling I got when I predicted you were eventually going to map things to a Klien bottle surface when you first mentioned the Chekov's gun bit was super cool!
Okay - be kind with me but there are some things , ideas really, I would like to bring to comments section about this video. They are meant positively and hopefully they will be taken positively.
1 - first there are not really 4 dimensions: x, y, d and theta. There is a mashup of 4 variables in 3 dimensions. Orthogonality took a dive somewhere. Alternatively look for growth in two directions simultaneously (see 3 below)
2 - there is no intersection problem in Klein's bottle. Taking a linear progression along [0, 1] we can put the "intersection" anywhere in that range.
Then the bottle can grow from anywhere in from zero to 1 and/or from 1 to zero. Preferably a distance away from the endpoints BUT the bottle grows in both directions simultaneously
Seeing it that way there are no problems at x=0 and/or x=1 otherwise at the endpoint x would simultaneously be zero and one. This allows x to grow increasingly from a closed zero in direction of 1 AND x to grow decreasingly from closed 1 in direction of closed zero.
The growths do not intersect they merely meet.
I suppose it could be argued that there has to be a join somewhere but as that would be infintessimally close at infinitely many points on a smooth surface who would notice the join?
3 - linear growth in one direction is fine but linear growth in two directions is even better 🙂
This video is genuinely amazing. I don't think any other video has made me truly understand a problem as much as this one.
I really like the way you keep me interested and in love with math even when life is trying to distract me from its beauty
Thank you 3B1B and Merry Christmas!
This 2nd edition to previous videos is very welcome. Great video once again. You never disappoint.
I'd LOVE to see a video on number theory
Yes please
Number theory is beautiful.
indeed, about time he leaves the riff-raff behind...
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.
I always thought that topology was the weirdest, somewhat useless (but at the very least still kinda neat) branch of mathematics until I started taking analysis classes. Suddenly topology seems like one of the single most important and fundamental branches of math.
Oh. Oh this is wonderful, I can't like immediately think of applications but the notion of mapping constraints on some set of values to a topological class that we can say things about dose seem useful.
Coming from engineering I think a lot about how we represent physical constraints like the shape and loading of materials or the limit of inputs to a control system.
Additional the, restricting a set of continuous points to a unit square and then showing things about sets of points in that space reminds me a lot of statistics and spesificslly statistical mechanics where things like particle location and interaction can be represented similarly.
This, this is good.
Topology has interesting and productive core intuitions, but as long as we try to speak of topology in terms of "points" and "neighbourhood", "point" in that language remains "undefined primitive notion" and as such antimathematical dishonesty, just like Weierstrass' "nearness" (which is regularly presented in scary quotes for the obvious reason that it is incoherent antimathematical blather), we can't solves problems caused by mathematical dishonesty by remaining foundationally dishonest and pretending that the Zeno-absurdity of point-reductionism can make any genuine mathematical sense.
Hint: to solve the rectangle problem, study origami. The most important topological evolutionary step after straight edge and compass.
18:50 wouldn’t flattening the mobius strip edge to the 2d circle just show the example of the infinitly close intersection for circle (i mean the dome in 7:47)?
This is definitely a contender for best video you’ve ever made
Thank you 3B1B, for releasing incredible and inspiring work for anyone with an Internet connection to learn from and enjoy !
I remember being presented with this argument when I was in 10th grade over 10 years ago and not liking it very much, because the fact of the non-embedding of the Klein bottle to me is as non-intuitive as the desired rectangle result itself. Speaking of which, the argument at 22:30 works only in the smooth category (you implicitly use tubular neighborhood etc), and although the fact itself is true for continuous embeddings, I don't think there's an elementary argument there to exclude wild embeddings (and quick googling leads me to the mathoverflow question which basically agrees with this assessment).
The mobius strip surface is the coordinate system of all possible non-unique point combinations. Each point on the strip represents a pair of points on the 2d shape. I thought the next step would be to plot "d" (the distance between those two points) as some "height" that is "above" the mobius strip. It seems like *that* surface (the "inflated mobius") is what the "weird architecture" surface maps to, and any spot on the "inflated mobius" where the heights are the same would represent two (x,y) pairs that generate an inscribed rectangle.
Amazing video.. i do however want to point out that you elude to the idea of homeomorphisms and suggest they are the continuous maps but as im sure you know they do have 2 additional constraints.. it is a creative decition on your part for sure but i wouldve loved to be mentioned that its not just continuity that gives this equivalence
This is one of the coolest videos I've seen in a long time, and I've forgotten most of the math I've ever learned.
One interesting idea could be to find a way to relate the rectangles in a circle to the ones on a given shape, then possibly you could find similar topological consistencies with the fractal shapes, like how does deforming a circle and its rectangles/square map to the snowflake
And with fractals to, if you can prove in the case of the snowflake that the first iteration has rectangles/squares, then the subsequent ones will most likely do to based on what parts are fractaling, likewise there probably a proof idea for infinite squares in self similar fractals, like the snowflake (i could see something like the Julius set possibly being harder, as it changes as you zoom in, where as the snowflake doesn’t)
New 3b1b video on Christmas Eve???? Best present ever, thanks Grant.
I sat through the entire video before saying "this isn't about topography at all."
"I'll talk about smoothness in a moment"
Oh I think spline continuity and smoothness has been embedded deep into my brain since the adventures I had tryibg to make a perfect procedural 3d train/rollercoaster track
This reminds me a lot of my PhD studying the mid-point Locus, symmetry sets, the rotational symmetry set. For the mid-point locus you take pairs of points on a curve, such that they both lie on a circle which is tangent to the curve at both points. In the mid-point locus for each such pair of points you plot the mid point. For the symmetry set, you take the center of the circle. In both cases the pairs of points have a local reflectional symmetry. The rotational symmetry set, which I studies captures local rotational symmetry. In all three cases the set of mid-points/centers creates a new curve, that in some sense capture the overall shape of the curve. While there may not be an application for inscribe squares there is a lot of interest in mid-point locus and symmetry set in the field of computer graphics.
My main task was writing code to calculate and draw these curve. Here I pretty much had to figure out the topology, a Möbius strip, like you have done, and then ensure algorithms worked across the joins.