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This open problem taught me what topology is [video]
- devil12gamer1 2y agoLove you
- Oarch 2y agoIt'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.
- Darthy 2y agoI'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.
- darthoctopus 2y agothis 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.
- Avshalom 2y agohe 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)
- deleted 2y ago[deleted]
- suryajena 2y agoGreat 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.
- Masterjun 2y agoHe 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.
- deleted 2y ago[deleted]
- raincole 2y ago> 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.
- mauricioc 2y agoThe 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 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.
- looneysquash 2y agoI'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.
- suryajena 2y agoThis 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.
- szundi 2y agoWhat if you print it? Your presentation is useless then /s
- yapyap 2y ago1 frame at a time
- formerly_proven 2y agoPDF supports embedded, interactively manipulatable 2D and 3D graphics/objects.
- atoav 2y agoThat 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?
- ykonstant 2y agoI've never bothered to look into a TeX based way to do this; is it something that can be done with TikZ/PGF?
- Qwertious 2y ago
- leoc 2y agoEh, 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.
- EdwardCoffin 2y agoI 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.
- throwaway98797 2y agouseful to smooth brains like me also meta lesson on how useful extra dimensions can be
- ykonstant 2y agoThis comment made me wonder if there is an analogous "inscribed cube" problem in three dimensions which is easier for smooth closed surfaces (≧▽≦)
- mettamage 2y agoI 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.
- scrollaway 2y agoThe 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").
- CT4u8798 2y agoI 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.
- chrsw 2y agoI 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.
- whatshisface 2y agoTime 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.
- CT4u8798 2y agoDonnie Darko style.
- philipov 2y agoTime 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.
- 2y ago
- bddg22 2y agoHaha 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.
- jebarker 2y agoI 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.
- noqc 2y agoThe 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.
- jebarker 2y agoI 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.
- noqc 2y agoyou actually said "I don't think I could have tied the esoteric word of topology to a practical problem".
- blueredmodern 2y agoIs there some area of math that you consider particularly useful for software developers?
- jebarker 2y agoDepending 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.
- WhitneyLand 2y agoDoes anyone else feel anxiety watching this? I guess some fear of failure/over achiever residual worry hangs on.
- klysm 2y agoAnxiety about not understanding immediately?
- WhitneyLand 2y agoYeah, weird right? It’s related to what’s sometimes called gifted kid burnout.
- klysm 2y agoYeah I can relate, high expectations result in disappointment eventually
- deleted 2y ago[deleted]
- boothby 2y agoWe generally don't talk about downvotes... but I gotta say that it's sad that this comment was gray. Your approach to an uncomfortable feeling was to name it, and get curious. That's commendable, and you're brave to share such in public. We shouldn't be punishing that here. Curiosity, like the response of klysm, is warranted. I've got a PhD in math and I've largely retreated from the academic pursuit. The thing that got me through my degree wasn't a drive for success or academic attainment, but love of the journey. Once I found employment, math turned dark and scary to me for quite some time, and this video was a breath of fresh air. I hope you find a source of joy that you can apply yourself to. From such a root, you can flourish. It needn't be work, in fact, I believe that the perilous job market underlies my anxiety. My root is my chosen family, not my career. With that security, it's easier to let one's mind wander and pursue puzzles like this open problem (should they capture you). But it starts with curiosity. Once, at a conference, John H. Conway admitted to me that he felt the very same as you for a period early in his career. And speaking of failure: I woke up with an idea for how to approach the open problem. I hacked up some code to apply my approach to the Koch snowflake. In writing it out, I found the obvious problem with my approach (context-free punchline: spotted the division by zero before I wrote down the line of code that would have triggered it). It was fun to fail, because nothing depended on me succeeding in that effort. And spotting bugs before they're written is always satisfying.
- madihaa 2y agoThis video has now taught me what topology is.
- klysm 2y ago3b1b 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
- shagie 2y agoThe 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.
- klysm 2y agoI 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!
- FLT8 2y ago> 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. This is true if Grant is the only person doing the work, however having a well educated and scientifically engaged populace seems important enough that we (the human race) should devote a few more resources to creating high quality (and freely available) courseware for all curricula/year levels.
- ninetyninenine 2y agoI’ve had classes where I didn’t want to learn shit but I learned anyway because of videos like this. Like the explanation is so clear that as long as you don’t fall asleep you absorb it. I didn’t become interested in science and math until later in my life and I spent much of my childhood in classes where I didn’t care.
- gcanyon 2y agoI’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.
- edanm 2y agoIf you haven't seen Dr Tadashi Tokieda's lectures on geometry, I highly encourage you to watch at least the first one. The best introduction to any math topic I've ever seen, I think, based around (among other things) the Mobius strip. https://www.youtube.com/watch?v=SXHHvoaSctc&list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ https://www.youtube.com/watch?v=SXHHvoaSctc&list=PLTBqohhFNB...
- gcanyon 2y agoThis is awesome, thanks!
- deleted 2y ago[deleted]
- graycat 2y agoAnother 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.
- S7UD27542 2y agoOK LEMME SEE
- kidneystereotyp 2y agoanytime i see math i wonder what it would be like to have a life and freedom to pursue
- derelicta 2y agoStill remains magic to me lol