動画数：214件
At you say "higher frequency light, like purple..." But no wavelength of light is "purple" (or more technically "magenta", since purple is just desaturated magenta in the same way that brown is desaturated orange). Magenta/purple is what we see when our eyes are simultaneously receiving blue light and red light, without much green light. It's a color that's invented in the brains of animals that have 3 color cones (like humans), which doesn't correspond to any one wavelength.
John
last year I got sucked pretty deeply 🥴
ok, it makes sense that if the two waves are in opposite phase they cancel out, but since the intensity of the wave decreases along its path, when we take a point on the wall which is not at the same distance from the two slids, the two waves meeting at that point will have different intensities, so they will not exactly cancel out even if their phases are perfectly opposite
- Inscribed squares
Actually I know the answer to the question. I'm not going to tell anyone the answer unless I get £1 billion no strings attached, non refundable, and you waive your right to sue.
"closed continuous curve" - immediately I started thinking:
Typo at , it says Closed *Continus* Curve
Isn't the (Open question) just a corollary of 1:1 aspect ratio rectangle)?
The animation here is wrong? The yellow rectangles in sime frames are obviously not squares...😅
unrelated to the question, but i love how the inscribed square freaks out and bounces around the curve as it morphs, its almost mezmorising
- Preface to the second edition
I truly hope they become more common, as a large portion of scientific videos which were accurate to how we understood things at one time, have become very dated rather quickly, especially in cases where the communicator grows in their understanding. I'd love to see more second editions of your videos!
having watched that one of the "earliest" video near the beginning of this channel I immediately recognized this and was wondering if it was a reupload. It is also one of the most memorable, as there's hardly math videos any on the implications and practical applications on topology, especially in a visual way.
through .... okay but what about 3:7, or 5:6, those combinations of points are completely ignored.... with this I hope you see that you have to also move the first endpoint to have a full mapping, fixing it in place is not enough
I challenge your assumption that second editions are less of a thing for YouTube compared to books. I think that concept is just as, if not more, important on YouTube. Worst case, you get some negative comments about "repeat" content. Best case: you get more views, keep us informed, and continue to show how math and education evolve.
(Just take
It's pm christmas eve for me...
read: "as well as humanly possible"
someone in the comments of your short actually mentioned, how this could be relevant! When we look a stable placement of a square table, it comes down to looking for such a square in the contour lines of the ground!
I have seen so many STEM related Youtube videos so that at I thought you were segueing into an ad for Brilliant :)
I was *sure* this was the lead up to an ad read for Brilliant or the like. The internet will make cynics of us all :P
I straight up skipped from to
: non-orientable is not the same as one-sided! In fact, not every Möbius strip is one sided! Being one-sided is a property of an embedding into some surrounding space and does not make sense for surfaces on their own.Möbius strips are always unorientable. Möbius strips in R^3 are always one-sided, but there are other three dimensional manifolds (think about 3D version of Klein bottle where you identify sides of a cube) in which you can embed both one-sided and two-sided möbius strips.
At , what is that mug, and how do I get it?
on pure instinct and had to go back when I realized that it wasn't a Brilliant sponsored segment at all
- The main surface
*@[****]:*Same midpoint guarantees point symmetry, point-symmetric quadrilaterals are automatically parallelograms.Same midpoint and same length guarantees 2-fold line symmetry on top of that, 2-fold line-symmetric parallelograms are automatically rectangles.
i dont get it. why do both endpoints have to move to find all possible combinations of "two endpoints"? wouldnt it be a simpler example to let just one endpoint move along the curve and only move the other one until we finished the whole curve?
Wouldn't this "function" be multi-valued? Why is it impossible to move the two points in such a way that their connecting line has the same midpoint? It seems obvious that the midline's length would change but the position of the midpoint wouldn't, giving a second value for the function.
this would map pair of points + distance into 5D spaceTo be precise, you are mapping distance between points + coordinates of the middle
,
Wow.. mind blown at the mapping on … what a solution masterpiece! Bravo…
Thank you for this video, I had an AHA moment where I understood the solution around the minute mark. The frustrating thing about mathematics for me is that I am only really interested in understanding these problems through their solutions, and not the other way around - I do not care for solving these problems myself to understand these things. It makes math very frustrating to study as a hobby, since I don't *do* mathematics at all.Well, I did code a fibonacci adder in python, but I lifted the logic of the code off a youtube vid. One can't exactly call that doing mathematics.
P.S. I haven’t finished the video yet, I’m only at so you may answer this question already, but I want to write it down before I forgot it just in case.
Does the surface have any hobbies? Is it a dog or cat surface? What are its life goals and aspirations?
Woah, you just read our mind
isn't that yellow rectangle animation missing a continuous rotation?
*@[****]:*Furthermore, due to the omni-directional symmetry of a circle, all inscribed parallelograms of a circle must be a rectangle.
someone please help! I've pondered for 3 days now but don't understand why the intersection is a line and not a point. You can't translate the rectangle sideways since it'll break its 90* angles. What am I missing?
*@[****]:* It probably has something to due with all inscribed parallelograms of a circle automatically classifying as a rectangle, due to a circle's omni-directional symmetry, but the same not being true for an ellipse.
is it because the length d of the pairs of lines in an ellipse gets shorter when u draw it from the flatter sides so they meet at a lower point in 3d space then as it becomes bigger the distance d increases thus mapping a whole vertical line segment
He basically shows you at . There are multiple pairs of varying lengths that go through the center point. Because these are mapped to different heights, it creates a vertical line.
There it is!
Why is the graph of the function of x and y shown in 3D and not in 2D? If the graph is in 3D, there should be a z variable? No?
good note there, I thought it was misqouted because the only version of this statement I keep hearing mentions a rifle, not a pistol. It's always a movie review too :)
The maths teacher I had in grade 10 in every exam included one task where there was superfluous information. It's genuinely good because in real life, you almost always have more data than is strictly necessary to solve something.
Should this be (x,x,0)? Later you say the surface is a Möbius-Strip. But the surface comes from adding height. I do not get that yes.
i love at when i realized that the seemingly "intersecting areas" are just points that have a mapping for multiple pairs. the fact that the mappings share the same output X and Y coordinate indicates that the pairs have the same center, and the fact that the outputs have the same Z coordinate indicate that the pairs are the same distance apart. and those two rules detaisl an inscribed rectangle
is hard to understand but I’m gonna keep pushing through
- The secret surface
At I was like “AAAAH ITS A MOBIUS STRIP AAAAAAAAAAAAAA”
To get a torus, you need to glue the opposite edges together (compare to ). At
Me at : Is it gonna make a Klein bottle? No it's a torus.
this made me finaly realize why grid cell activity can be represented as a point on a torus!
Yea, but the Klein bottle can also be represented as a donut shape as we saw at
I think I actually heard my brain click at ; I can't gush about this channel enough.Bravo!
hence proved earth is a donut. 😂
Me seeing the torus: 3B1B when I catch youuuuuuu
Encoding pairs of points on a loop as points on a torus is so simple and so smart
so in theory you could unwrap a torus and get something that perfectly tiles.
oh wow, you can define n-dimentional toruses as n-number of points on a closed loop. Cool!
Wiggle a bit. Wiggle it a little bit. (The Kiffness)
At I got so excited because it all clicked together for me. I literally shouted "that's why we get a Klein bottle!"
I first came across this method of isolating unordered pairs in Physics for the Birds’ “Why Möbius Strips Make Better Pianos.” (Side note: Physics for the Birds is easily the most underrated channel on this site, imo, and I think that anyone who enjoys 3Blue1Brown would also get a lot out of his videos.) So you can imagine my surprise when, at
Thank you, this is great. I would love if you explained with the Torus still in the picture. I want to see how that transforms into a Mobius strip.
Me at : Is it gonna make a Klein bottle? No it's a Mobius strip.
I got lost on the step at 😅Until that point, the folding of the mapped square into a torus or a right angled traingle made sense because I thought of these operations as a continuous transformation of the whole 2D plane/the axes itself {and they also concerned a reason like x=0 & x=1 and y=0 & y=1 lines are the same, and points (a,b) = (b,a)}. But the sudden cutting of the right angled triangle (and the vanishing of the axes) seems hard for me to wrap my head around. Is there an intuitive reason behind it?
gives you this spine shivering "I see where we are heading"😅
dividing the square in half to remove the redundant points makes sense. But re-arranging it into a mobius strip, which is non-orientable, blew my mind. Unordered pairs of points on loop=mobius strip
I literally laughed out loud at when I realized it looked like we were gonna have a Mobius strip and then we did.
was that come around moment for me to fall in love with topology
mind=blown
Ok, I am lost at this point in the steps.
- Klein bottles
The diagram at reminds me of the First Isomorphism Theorem diagram I learned in my group theory course. I wonder if the argument structure in the proof is similar.
He never explained why self intersection is necessary to prove this. Can someone please help me to understand how it proves it?
Of course, the proof of the theorem is not easy and takes a lot of care. However, this theorem is used basically all the time in knot theory, so for someone familiar with the area it gives an easy way to see why the claim at is false.
Thanx, Dan Azimov, whoever you are. This is the most beautiful mollusk shell in the universe. I now need to build this thing and live in it, and redefine myself as a mollusk, as I most certainly do not feel human!!
If you watch closely it does intersect itself in a weird way
We love you, Grant, but I confess I felt a bit vindicated that you finally told us about a construct you felt trippy about and kinda hard to visualize 😆
Can someone please make this into a Rocket League map
mobius strips strictly have no “interior” or exterior tho?
What does it mean when you say "the interior of the mobius strip" at . Does it just mean the surface of the strip; since mobius strips only have one side?
If you glue two surfaces of the möbius strip and the reflected one together, would you be able to make a rectangular prism?
sticky fingers, paper cuts, and a head ache?... Lmao
Me at : Is it gonna make a Klein bottle? Yessss finally 🤣
Please somneone explain! At we already have a surface with two gluable edges to form a torus. Why do we cut it to make the klein bottle? What is the reasoning? Any surface can be contorted if you cut enough.
we are gluing the neighbouring edges
this was the second I realized the Klein bottle was comming into play and my jaw dropped. We'll done 3b1b
Guys... I don't feel so good... This might be a Klein bottle jumpscare...
Actually, one may not need the argument of Klein bottles at . if you consider the base (z=0) as a flat disc, this disc together with the upper complex surface form a closed surface. In topology, sewing a Möbius strip to the edge of a disc would create a closed surface called Roman surface. It indeed does self-intersect. Since the base disc is not self-intersect, the upper part must self-intersect.
I love the realization at thinking, "this isn't going to work... wait, you could make a Klein bottle!' and then realizing that the most famous property of Klein bottles (the fact they self-intersect) literally solves the problem
When you started morphing the shape into a Klein bottle at , it completely blew my mind because I was imagining bizarre twists of the shape to get the yellow arrows to point in the same direction so I did _not_ see the Klein bottle coming.
I had to pause and just sit for a while when the klein bottle was constructed , beautiful
at , why can't the self-intersection be at the corresponding points on the top and the bottom half, which does not give the desired distinct pair of points in the original problem?
Similarly, the proof at : works in R^3, but not general 3d manifolds.
But how does the argument break down in higher dimensions?
D the argument at must fail somewhere in 4D but I can’t figure out where.Does anyone know?
- Why are squares harder?
Edit: Just got to , looks like I was on the right track.
And at Grant - the master storyteller that he is - did of course get back to the nature of the curve and it did turn out that the proof was hard only for the "corner case" curves.
question: how did you numerically find all of these examples for this specific curve?
what would be a poorly defined tangent line?
wouldn't the fact that smooth curves can get arbitrarily close to rough curves also prove that it is true for rough curves? Any one smooth curve isn't the fractal, but in the limit it would be the fractal, and since at every closer iteration it maintains that inscribed square property, the rough curve must also? This isn't some gotcha, I'm sure people who've pursued this problem much more extensively than me have already ruled out why this doesn't prove it, but what is the reason this wouldn't prove it?
- What is topology?
It’s really frustrating to watch a video on topology with a lot of continuity arguments and always hear about the “wiggle”! Not one mention of preimage yet
this reminds me of the fractal video. How it was thought as a tool to encode roughness but became a kind of mathematical toy.
"sometimes in the process, the original problem-solving process gets lost" NO SHIT!!! Like, quadratic equations? Or, algebra? Or negative numbers? I still don't know what any of that was meant to solve in mathematical history.
Do 2 equivalent surfaces have the same surface area? It feels like that cuz every point on a surface corresponds to a unique point present on another but it doesn't look like they have equal surface areas
@ This is just about music theory semantics, but if I understood this part correctly, you would end up describing every possible pitch class pair, not every possible musical interval. I suppose the intervals could be described by using all real numbers from 0 to positive infinity?
Could you elaborate on what is the association between points on a loop and musical intervals? (mentioned at )
As a composer and a math enthusiast, this here scratched an itch I didn't know i had! I'd be really curious to understand how that would be helpful to describe every possible musical interval! Can please someone help me on this ? Anyway, cheers for the awesome video, as always!
, you brought up musical intervals as another example of a Möbius strip. I don’t think I’m alone in that finding deep or useful connections between seemingly unrelated phenomena is by far my favorite part of math and physics.
So, any musical interval is a point on a Möbius strip. Suddenly I realized why music is so limitless and beautiful.
bro just casually throws out "music is a mobius strip" as if those 5 seconds wouldn't compulsively obsess me with that concept for the next week!!
Bach is smiling with his crab canon right now.
"the reason that mathematicians get really excited about bizarre properties and impossibilities is not just aesthetic. it's because when you're looking for logical proofs, constraints and impossibilities are your fuel for progress"woah