478 points | by surprisetalk1 day ago
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 also miss my old subjects, not to mention being young and in university)
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.
Maybe this can aid in your learning.
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.
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.
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.
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.
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.
It was his original video on this topic which had me instantly hooked to 3B1B all those years ago.
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.
https://www.youtube.com/watch?v=SXHHvoaSctc&list=PLTBqohhFNB...
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
I completely disagree. Animations can be appropriate, but people have formed dogmatic generalizations due to shitty use of gifs
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.
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?
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
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.
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.
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.
also meta lesson on how useful extra dimensions can be