I loved this. I did my PhD in algebraic topology, but studied lots of topology so was familiar with this material. I doubt I could ever have explained these concepts so clearly or tied the esoteric world of topology to a "practical" problem.
Since my PhD I've had a couple of careers and ended up as a research software engineer working on AI. I often feel nostalgic about pure math (maybe even a little regretful I left academic math). But I think it'd be almost impossible for me to return to academic math. The 3B1B videos always remind me that math is available to all and you don't have to be a working mathematician to enjoy, learn, and even discover, new math. You don't have to be employed as a mathematician in a university.
I agree. My PhD is technically in CS but it made heavy use of algebraic topology. Being 5 years out, having worked briefly in tech, then at a national lab as a software engineer has given me enough outsiders' perspective on pure math. You probably need to work as a professional mathematician to be at the research frontier of a given area, but otherwise the fundamentals of math are unchanging, and in my opinion, that makes it accessible to anyone who is sufficiently interested in and passionate about math.
> working on AI
I think we're about to enter an incredible new age of mathematics, driven by AI and theorem provers. It's going to be hugely disruptive to mathematics, but lots of fun to amateur mathematicians.
Yeah, I really hope so. I'm hoping that my background is going to allow me to work/play in this area. I'm currently learning about theorem provers so I can get involved.
Recently on HN: https://news.ycombinator.com/item?id=42440016
Maybe this can aid in your learning.
This is great - thanks for sharing
Is there some area of math that you consider particularly useful for software developers?
Depending on the area of software development then trigonometry, geometry, linear algebra, number theory, combinatorics and probability theory are the most obviously useful. Beyond that I know that there's a close relationship between category theory and functional programming. I'm not familiar with the details of that or whether it's useful in practice or more of an area of theoretical study. I'm sure there's others on HN that know though. Interestingly I used a fair amount of category theory in algebraic topology, but never closed the loop and learned much about the relationship to programming.
The original understanding of a manifold was simply a "configuration space", which is very concrete, so I'm not sure what you mean that you are surprised that the world of topology could be practical.
I didn't say I'm surprised the world of topology could be practical. I said that _I_ wouldn't have been able to explain the concepts in the video so clearly and tie them to a practical problem.
3b1b shows us what’s possible in math pedagogy. I’m excited for the future of the space, but sad it will take so long to adopt methods like this for teaching math
The amount of effort to do a single 30 minute video of this sort when scaled out to a half or full year math class is significant.
Another consideration is that we learn things from it because we want to learn it. We are engaged with the topic the instant we hit play because we want to watch it.
Compare that with a high school or college setting where the majority of the class is taking it because they have to - not because they want to. This means that there's no initial engagement and a professor can't call out the student in the 3rd row from the back that is starting to fall asleep.
This can work really well for the people who want to learn it. However, it potentially adds to people who don't want to become competent in the material falling further behind.
I agree you can’t get around people fundamentally not being interested in the material. That being said, I still think that the power of 3b1b does should not be understated. It can cultivate interest as well!
There is no royal road to geometry just like the route to Carnegie Hall is practice practice practice.
I’ve known about the möbius strip since I was a kid, and the idea of existence proofs based on continuous functions having to cross since my early teens.
The idea that the möbiu strip is more than a pointless novelty has never occurred to me, and now I feel like I have to apologize to that object for dismissing it so cavalierly. Its role in this proof is remarkable and a wonderful brain tickle.
It's great that he revisited this problem.
It was his original video on this topic which had me instantly hooked to 3B1B all those years ago.
I have no clue about maths beyond extremely basic stuff, but am fascinated by this sort of thing, and I need pictures to understand stuff like this. What an excellent video. During it, when they introduced how you can map the 2D to 3 dimensions, my initial thought was "I wonder if this is how you could map 3D into the 4th dimension?". Then later they mentioned 4 dimensions. This is something I cannot visualise or really understand.
> I have no clue about maths beyond extremely basic stuff, but am fascinated by this sort of thing, and I need pictures to understand stuff like this.
Fascination is all you need. I find many people have a lot of self-limiting beliefs around math. There’s many reasons for them to develop, but I firmly believe that many people are legitimately interested in mathematics and have the capability despite their beliefs.
One of the problems with math, like a lot of things, is that even though you may find it deeply interesting and fascinating and you may even see great utility in it, becoming an expert is very difficult and is fraught with a lot of failure which many people can't, or won't, stomach.
I gave up on trying to visualize 4 dimensions. I don't know if it's possible. Instead I just try to think of 4D as more of ideas and less geometry: rules, consequences, capabilities, etc. We can do the same thing in 3 dimensions by saying things like "two objects can't exist at the same place and at the same time" or "parallel lines meet at infinity" or "parallel lines never meet" or something. We usually don't do that for 3 dimensions because we have visualizations and intuitions which we can use instead of breaking everything down formally all the time.
I've always wanted to make a 4d space in VR. That way it's only one dimension higher, technically. Could help to visualize it in a way that hasn't been done yet
There’s a video from the same channel on visualizing quaternions as a projection into 3d that was really fun for this. Only a restricted section of a 4d space, but i feel like the principle generalizes a little because of the idea of, like, imagining one 3d space thats finite as equivalent to an infinite 3d space, just stretched
Time is nature's forth dimension, so I think considering the various stages of a slice moving through a four dimensional object at once counts as a visualization.
Time is not a dimension of the same kind as spatial dimensions. It has a different metric and you can’t move freely back and forth on it. When you rotate on the XT plane, it doesn’t mean the same thing as rotating on the XY plane. It is not a good candidate for the sort of fourth dimension we’re interested in.
The Euclidian group in four dimensions is a fourth dimension, but the Lorentz/Poincare group is the fourth dimension. ;)
Donnie Darko style.
Haha great to see him mentioned: Lobb taught my Linear Algebra 1 course a few (god im old) years ago. Excellent prof, and we still laugh over the looks of despair he gave us when we didn't get something.
I've got a problem with that video that starts at 4:15. He seems to jump to the conclusion that for every midpoint there is only 1 distance. But that midpoint is formed by picking 2 points on the edge, and one could easily pick two other points on the edge that have the same midpoint (but have a different distance). He did not address that point at that point in the video, and for the next 2 minutes I kept raising that point in my mind. After he continued down that path not addressing that point, I felt that I must have missed something, or that more intelligent math viewers would have solved that open question in the mind in seconds and I am not mathematically inclined enough to be the target audience. And I stopped watching that video.
I think good educational videos are the result of a process where a trial audience raises such points and the video gets constantly refined, so that the end video is even good for people who question every point.
He addresses this at 9:00 in the video. You're thinking of a function graph, but he never made a function. He just sets up a visualization of a set of 3D points.
The function defined in the video is "Given a pair of points A and B on the curve, output (x, y, z), where (x, y) is the midpoint and z is the length of the segment connecting A and B", and the pictures are of its image, not its graph. But if you define it visually, then it's very natural to misunderstand it the way you did, since the picture looks a lot like a function graph of a function which takes midpoints (instead of pairs of points) and returns the distance corresponding to that midpoint (which is not well-defined, as you pointed out). If this happens, the viewer is then completely lost, since the rest of the video is dedicated to explaining that the domain of this function is a Möbius strip when you consider it to consist of unordered pairs of points {A, B} (as one should).
Ultimately, if you don't have a 100% formal version of a given statement, some people will find a interpretation different from the intended one (and this is independent of how clever the audience is!). I think 3Blue1Brown knows this and is experimenting with alternate formats; the video is also available as an interactive blog post (https://www.3blue1brown.com/lessons/inscribed-rect-v2) which explicitly defines the function as "f(A, B) = (x, y, z)" and explains what the variables are.
The fact that "given a large enough audience (even of very smart people), there will be different interpretations of any given informal explanation" is a key challenge in teaching mathematics, since it is very unpredictable. In interactive contexts it is possible to interrupt a lecture and ask questions, but it still provides an incentive to focus on formalism, which can leave less time for explaining visualizations and intuition.
this is not a conclusion that he jumps to! all that is stated is that there is a mapping from every pair of points on a curve to a set of 3D coordinates specified by their midpoints and distances. there is no requirement for uniqueness here. in fact, the whole point of this is to turn the search for an inscribed rectangle into the search for two pairs of points on the curve that have the same midpoint and distance --- this is stated just 1 min 15 seconds after the timestamp that you point out.
> I think good educational videos are the result of a process where a trial audience raises such points and the video gets constantly refined, so that the end video is even good for people who question every point.
It would be at least as long as a one-semester course in typical math major then.
To address your specific question: he doesn't assume each midpoint has only one distance at all. He doesn't say it and the visualization doesn't show it as so.
he maps two points (by using their midpoint) and a distance to the (x,y,foo) if it was two different points with the same midpoint but different distance it would map to (x,y,bar)
I'm don't feel like I really get the distinction between a mapping and a function, or a visualization and a graph.
But he was careful to point out that it wasn't a graph.
To me the key point is that the input is all three variables, the two points and their midpoint, and not just the midpoint.
Great point now we can raise the issue and he will do a revision 3, with even better explanation for those issues just like in the books.
This video has now taught me what topology is.
Another view of topology is in
John L.\ Kelley, {\it General Topology,\/} D.\ Van Nostrand, Princeton, 1955.\ \
In the set R of the real numbers and x, y in R with x < y,
(x,y) = { z | x < z < y }
is open and, with x <= y,
[x,y] = { z | x <= z <= y }
is closed.
A subset of R that is both closed and bounded is compact, a powerful property, e.g., in Riemann integration.
And so forth but in topological spaces much more general than the real line and open and closed intervals. Apparently hence the "General" in the book.
As a math major senior in college, read Kelley and gave lectures to a prof. But now there are some other definitions of topology.
Does anyone else feel anxiety watching this? I guess some fear of failure/over achiever residual worry hangs on.
Anxiety about not understanding immediately?
Yeah, weird right? It’s related to what’s sometimes called gifted kid burnout.
Yeah I can relate, high expectations result in disappointment eventually
This video if it was a scientific paper I would have visualised absolutely nothing. I don't know that if we can submit/embed animations instead of PDFs for university classroom work/ scientific papers, because that's really much better than having to read papers/PDFs that is so incomplete without the right imagination/visualization of the problem. The last time I was giving a mock seminar in my university using a GIF to explain the RRT algorithm I was warned to not use animations in presentations . . . I mean either it was really not that helpful to visualise the solution or it has to do something with age old standards that needs to be revised. I mean figures can only do 3 or 4 frames isn't more frames better.
If you need to visualize an algorithm in a talk, the usual approach is having a few slides representing the key steps instead of an actual animation. That way you can adapt the pace to the audience, stop to answer questions about any individual frame, and jump back to previous frames when necessary. People often find animations on slides distracting, and the forced pace is almost certainly wrong. And if the animation is longer than a few seconds, the talk stops being a talk and becomes an awkward video presentation instead.
> People often find animations on slides distracting, and the forced pace is almost certainly wrong.
I completely disagree. Animations can be appropriate, but people have formed dogmatic generalizations due to shitty use of gifs
Careful where you say that :-)
In 2002, when I was doing my second year at college, my professor was cool enough to let me submit an animation of the self-balancing insertion algorithm for AVL trees. Those were the years of Macromedia Flash and Director. It was a cool project, and I wish I had kept the files. Overall, it was a highly technical thing.
Twenty and so years later, I still do animations, even if only as a hobby. These days I use Blender, Houdini, and my own Python scripts and node systems, and my purpose is purely artistic. Something that is as true today as it was twenty years ago is that computer animation remains highly technical. If an artist wants to animate some dude moving around, they will need to understand coordinate systems, rotations, directed acyclic graphs and things like that. Plus a big bunch of specific DCC concepts and idiosyncrasies. The trade is such that one may end up having to implement their own computational geometry algorithms. Those in turn require a good understanding of general data structures and algorithms, and of floating point math and when to upgrade it or ditch it and switch to exact fractions. Topology too becomes a tool for certain needs; for example, one may want to animate the surface of a lake and find out that a mapping from 3D to 2D and back is a very handy tool[^1].
I daresay that creating a Word or even a Latex document with some (or a lot of) formulas remains easier. But if I were the director of a school and a student expressed that videos are easier to understand, I would use it as an excuse to force everybody to learn the computer animation craft.
[^1]: Of course it's also possible to do animations by simply drawing everything by hand in two dimensions, but that requires its own set of skills and talent, and it is extremely labor-intensive. It's also possible to use AI, but getting AI to create a good, coherent and consistent animation is still an open problem.
PDF supports embedded, interactively manipulatable 2D and 3D graphics/objects.
That word "support" does a lot of heavy lifting there. A bit like in "Email supports end to end encryption".
You are not wrong, but if you had to bet your life on somebody being able to get the information and you don't know how they are going to view that PDF would you do it?
I've never bothered to look into a TeX based way to do this; is it something that can be done with TikZ/PGF?
What if you print it? Your presentation is useless then /s
1 frame at a time
Eh, it’s charming but honestly, for someone who doesn’t already know the maths it’s still just edutainment, as it leaps from trivial to incomprehensible in the blink of an eye.
I don't think he's trying to make people understand the proof, rather to show them that topology really has an application for problems that aren't themselves topological in nature, and it is comprehensible enough for that purpose.
Well that's the "edu" part of edutainment. Sometimes you've gotta rewind or pause and think about what's being said to make sense of it. I do understand that sometimes videos go way too fast and leave tons of stuff out and that's very frustrating but 3b1b is a pillar of the community for very careful and complete descriptions of things. But then also the "tainment" part would signal that there's no need to watch if you're not interested.
But all this could be my bias of having some math background, though never having studied topology or even analysis from anything like a class or textbook. Felt like the video was aimed directly at people like me
I disagree, I'm not well-versed in math but I felt I could follow most of it.
What I don't get though is the jump from the mobius strip to the klein bottle.
He just goes and does it and duplicates the surface to reflect it to the original one. I do understand to some extent that once you have to assume the klein bottle is the shape you're looking for that because it's self intersecting, it must mean that you have 2 different points on that same surface and therefore 2 lines of equal length with the same midpoint.
The point of the jump is that if you want to track an extra coordinate and visualize it with the restrictions he mentions, then the klein bottle is the correct topology (the correct "visualization").
Oh, so you mean to say that:
1. The positive surface is for tracking one midpoint for coordinates A and B
2. The negative surface is for tracking another midpoint for coordinates C and D
Together it's a klein bottle. Klein bottle's always intersect, so therefore there's always an intersection of the two midpoints, which is why there's a set of points A, B, C and D such that line segments A and B are equally long as C and D going through the same midpoint.
The surface with the interior of the loop added forms something called a projective plane. A Klein bottle is just two projective planes glued together. Neither can be embedded in R^3 without intersections.
In many ways, I agree. I have an engineer's understanding of math for my discipline but topology is most definitely not one of them. Through his graphics, I could most follow the gist of what he was attempting to get across but when it was over, I honestly had to ask my self, what did I just watch. Perhaps watching it again, really concentrating on it, and trying to understand, might help, but, in reality, it is so far out of my interest zone, I'll never do it.
useful to smooth brains like me
also meta lesson on how useful extra dimensions can be
This comment made me wonder if there is an analogous "inscribed cube" problem in three dimensions which is easier for smooth closed surfaces (≧▽≦)