This open problem taught me what topology is [video](youtube.com)
youtube.com
This open problem taught me what topology is [video]
https://www.youtube.com/watch?v=IQqtsm-bBRU
87 comments
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.
When I was doing my degree (area studies and linguistics), a friend who was in mathematics liked to tell me that mathematics was the second-most democratic science: all you need is a pen, some paper, and a waste paper basket; the humanities were the only thing that was more accessible - since we don't even need the waster paper basket...
(I also miss my old subjects, not to mention being young and in university)
(I also miss my old subjects, not to mention being young and in university)
> 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.
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.
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.
you actually said "I don't think I could have tied the esoteric word of topology 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.
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.
> 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.
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.
High school math has a standardized curriculum that doesn't change significantly - it's 100% possible to create this high quality material for the whole n years and make use of it year after year. Especially that the most important part of this is the software used to make these semi-interactive graphics (which is open-source), so a teacher can just do it their own way, incorporating animations fit for their examples - no need to pre-render a video for each day. Do a "normal" class and visualize important aspects.
I’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.
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.
I usually watch 3b1b without any prior “want to” or any idea what it will be about. For me it’s the format that drives interest.
Although I’m from a natural-math-guy group, in a sense that I usually have no issues with understanding the material, in contrast to these “so interesting, but I understand nothing” comments below it. I always wonder why they watch it, cause it must be just a set of vector animations then.
Although I’m from a natural-math-guy group, in a sense that I usually have no issues with understanding the material, in contrast to these “so interesting, but I understand nothing” comments below it. I always wonder why they watch it, cause it must be just a set of vector animations then.
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!
Yes and no...
Ultimately it would be impossible to come up with these greatly simplified explanations without the complexities and notation taught in the pedagogy you are dismissing.
Granted though, as students, the gifted ones often already have this picture in their mind, so the intuition is obvious. So to bring those less gifted or familiar with a topic up to speed, things like this make a ton of sense.
Ultimately it would be impossible to come up with these greatly simplified explanations without the complexities and notation taught in the pedagogy you are dismissing.
Granted though, as students, the gifted ones often already have this picture in their mind, so the intuition is obvious. So to bring those less gifted or familiar with a topic up to speed, things like this make a ton of sense.
There is no royal road to geometry just like the route to Carnegie Hall is practice practice practice.
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.
It was his original video on this topic which had me instantly hooked to 3B1B all those years ago.
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.
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.
If 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=PLTBqohhFNB...
https://www.youtube.com/watch?v=SXHHvoaSctc&list=PLTBqohhFNB...
This is awesome, thanks!
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.
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 guess that's true for most things. Say, learning to play an instrument can be similarly difficult at first.
Motivation is vaning, you need discipline to actually stick to something and get better at it. But even getting better day-by-day by only a tiny percentage will result in huge gains over long periods.
Motivation is vaning, you need discipline to actually stick to something and get better at it. But even getting better day-by-day by only a tiny percentage will result in huge gains over long periods.
I believe the most important aspect for learning math is being comfortable with not understanding and being able to smack your head into the wall repeatedly. It takes some stubborn determination to push past
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 4D mini golf VR game you might be interested in checking out. It's called, uh, 4D Golf. Creative! I've not played it myself, but it's on my list. I hear it's pretty cool!
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.
My understanding is that time can be a 4th dimension, but n-dimensional spaces themselves are simply a very basic mathematical structure, where a point can be described by n numbers (you can actually be abstract even in that, no need to stick to rational numbers, I believe).
As long as you can map time to a number line, it's a valid representation. We just happen to have hardware acceleration for 3-dimensions, and the 4th is just completely unintuitive to us.
As long as you can map time to a number line, it's a valid representation. We just happen to have hardware acceleration for 3-dimensions, and the 4th is just completely unintuitive to us.
If we're only talking about simple vector spaces, your understanding is accurate, but when we're talking about visualizing shapes in 4 dimensions, we typically want something more. We are doing geometry then, and so we want a metric space that defines a concept of distance (which vector space don't have).
When it comes to geometry and not just vector spaces, time dimensions have a different definition of distance than do space dimensions. There's a minus in the formula where you would usually have a plus. And this means that shapes in this space behave very differently than what we're after when imagining a hypercube or hypersphere, for example.
We want to think of a 4 dimensional space where all the dimensions are indistinguishable, but the minus sign in the metric distinctly identifies the time dimension. For this reason, physicists typically call this kind of space a 3+1 dimensional space rather than a 4 dimensional one.
https://www.youtube.com/watch?v=GkCWywO93b8
When it comes to geometry and not just vector spaces, time dimensions have a different definition of distance than do space dimensions. There's a minus in the formula where you would usually have a plus. And this means that shapes in this space behave very differently than what we're after when imagining a hypercube or hypersphere, for example.
We want to think of a 4 dimensional space where all the dimensions are indistinguishable, but the minus sign in the metric distinctly identifies the time dimension. For this reason, physicists typically call this kind of space a 3+1 dimensional space rather than a 4 dimensional one.
https://www.youtube.com/watch?v=GkCWywO93b8
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.
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.
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.
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.
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.
> 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.
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)
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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.
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.
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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.
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.
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.
This video has now taught me what topology is.
Does anyone else feel anxiety watching this? I guess some fear of failure/over achiever residual worry hangs on.
We 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.
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.
Thanks so much for the thoughtful comment. I hope others find it as hopeful and motivating as I did.
Anxiety about not understanding immediately?
Yeah, weird right? It’s related to what’s sometimes called gifted kid burnout.
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OK LEMME SEE
Love you
Still remains magic to me lol
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
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.
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?
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?
PDF supports being printed out onto paper.
anytime i see math i wonder what it would be like to have a life and freedom to pursue
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
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.
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.
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 "positive" surface already contains all the necessary points. It's hard to prove that this surface on its own intersects with itself, but turning it into a Klein bottle makes the proof easy, since it's already known that the Klein bottle must intersect with itself when embedded in 3-D space.
It takes some rigor to ensure that mirroring the surface and turning it into a Klein bottle doesn't introduce a problem that would invalidate the proof, but the idea is this:
1) The surface exists only in the "positive" area above the x-y plane, and the mirror exists only in the "negative" area below the x-y plane.
2) The two surfaces only share the points on the original curve (on the x-y plane), and these points correspond only to the trivial cases where A=B. The surface and its mirror don't intersect anywhere else.
3) The resulting combined surface is a Klein bottle in 3-D space, which must intersect somewhere. Because of 2), that intersection must either be in the positive space or the negative space. Either way, that means there is an intersection in the original surface.
As briefly mentioned in the video, it's critical that the original constructed surface is only in the positive area, because otherwise when you mirror it and then turn it into a Klein bottle, the required intersection might just be the surface intersecting with the mirror, and not within the original surface itself.
It takes some rigor to ensure that mirroring the surface and turning it into a Klein bottle doesn't introduce a problem that would invalidate the proof, but the idea is this:
1) The surface exists only in the "positive" area above the x-y plane, and the mirror exists only in the "negative" area below the x-y plane.
2) The two surfaces only share the points on the original curve (on the x-y plane), and these points correspond only to the trivial cases where A=B. The surface and its mirror don't intersect anywhere else.
3) The resulting combined surface is a Klein bottle in 3-D space, which must intersect somewhere. Because of 2), that intersection must either be in the positive space or the negative space. Either way, that means there is an intersection in the original surface.
As briefly mentioned in the video, it's critical that the original constructed surface is only in the positive area, because otherwise when you mirror it and then turn it into a Klein bottle, the required intersection might just be the surface intersecting with the mirror, and not within the original surface itself.
> The "positive" surface already contains all the necessary points.
If this is the case then why are you allowed to duplicate the surface again on the "negative" plane? To me that gives the idea that you duplicate the whole original curve. He didn't motivate that step.
Or oh wait. I think I see it, a bit.
It's not so much duplication it's simply that by doing a negative visualization, you visualize it in a different way. But also in a way that relates to each other as the x,z plane are the same and the vertical (y) planes are inverted.
I guess one could say that the positive plane is for one midpoint and the negative plane for the other midpoint as the surface areas on those planes are a visualization of midpoints.
And then when you turn that into a klein bottle, you show how both midpoints relate.
If my understanding is correct enough, then I have to say, this is wild.
Thanks for explaining! This is really cool. Like I said, I'm not well-versed in math. To even have an understanding of following the main beats of it is really mindbending as most of it is new.
If this is the case then why are you allowed to duplicate the surface again on the "negative" plane? To me that gives the idea that you duplicate the whole original curve. He didn't motivate that step.
Or oh wait. I think I see it, a bit.
It's not so much duplication it's simply that by doing a negative visualization, you visualize it in a different way. But also in a way that relates to each other as the x,z plane are the same and the vertical (y) planes are inverted.
I guess one could say that the positive plane is for one midpoint and the negative plane for the other midpoint as the surface areas on those planes are a visualization of midpoints.
And then when you turn that into a klein bottle, you show how both midpoints relate.
If my understanding is correct enough, then I have to say, this is wild.
Thanks for explaining! This is really cool. Like I said, I'm not well-versed in math. To even have an understanding of following the main beats of it is really mindbending as most of it is new.
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
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 (≧▽≦)
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.