Dirac's belt trick, Topology, and Spin ½ particles
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- เผยแพร่เมื่อ 2 มิ.ย. 2024
- ANSWERS TO FREQUENTLY ASKED QUESTIONS:
scholar.harvard.edu/files/noa...
This is my submission to 3Blue1Brown's "Summer of Math Exposition 1" #SoME1. In this video, I explain what Dirac's famous belt trick has to do with the topology of rotating spin 1/2 particles, such as electrons.
I created the 3D animations using Three.js/CCapture.js, and the math animations with Manim Community v0.8.0.
00:00 Introduction
4:14 The space of rotations
9:40 Paths through the space of rotations
18:48 Group theory & the fundamental group
31:30 Quantum spin and SU(2)
39:31 SU(2) as the double cover of SO(3)
48:26 Bringing it all together
52:22 Tying up loose ends
Music by Vincent Rubinetti
Download the music on Bandcamp:
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Stream the music on Spotify:
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Largo From Concerto No 5
Exzel Music Publishing (freemusicpublicdomain.com)
Licensed under Creative Commons: By Attribution 3.0
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![[TH] 2024 PMSL SEA GF D3 | Summer | มาร่วมเป็นพยาน กับวินาทีแชมป์!](http://i.ytimg.com/vi/JCYvEsFlwx8/mqdefault.jpg)
Fun fact: All USB ports before USB-C have spin 1/2
I concur. First you try and it dos not fit. Then you rotate Pi and it still dos not fit. Then you rotate it Pi again and now it fits.
Wait, it'd be spin 2/3, right? cause you need 3 180º turns to arrive at the initial position
Also when you don't observe it it goes into superposition, and when observed it always collapses into the state where you need to rotate twice
USB Plugs were spin 2/3. PBS Space Time did this bit: th-cam.com/video/dw1sekg6SUY/w-d-xo.html
Yes definitely, although I got a USB a to micro port that is spin one on the a side like usb-c
A word of advice: if you discover a new phenomenon and immediately reach for a matrix representation to describe it, you should call your doctor and ask if the automorphism group of a vector space is right for you. Clifford Algebras provide a safe, sanitary, and intuitive alternative, and is recommended by nine out of ten dentists.
I was going to mention this! Especially in the beginning when he rotated about an axis rather than within a plane. Can we all please rotate in planes from now on? :)
This is pure genius.
@@nicolaihaasgiedraitis4082 jesus it was disgusting to see
@@nicolaihaasgiedraitis4082 hey, I know you wrote this comment 7 months ago, but I'm curious if you can tell me the timestamp of the part you're talking about. Because I'm curious about what you mean by "rotating in a plane".
Edit: I just looked it up and I found that specifying the plane of rotation is useful in higher dimensions (>4) and of course, you can't rotate about an axis in 4D or higher. But in 3D, specifying the plane of rotation is equivalent to specifying an axis of rotation, which is just a vector in the 1D subspace orthogonal to the plane. So I'm not sure what the problem is with talking about an axis of rotation for 3D. Unless I misunderstood your comment (still curious about the timestamp)
Too much work, can I hire someone to understand this?
"One other thing I wanted to mention is the Hopf Fibration. There. I've mentioned it." My sides. Twin Peaks reference much appreciated too.
I explained the belt trick to my class now everybody knows the colour of my underware
"Wait, a 360-degree rotation is a 360-degree rotation, you can't say it's only a half!" "Well, Albert """Henry""" Einstein..."
One of my favorite references, glad someone else got it :D
I can't believe a speedrunning meme got so ubiquitous as to show up in a video proof of a quantum-mechanical phenomenon. I love it
A man of culture!
It's funny that this is the SECOND time I saw this reference in the #SoME1 playlist
@@viliml2763
@Owen Maitzen is also a man of culture!
Oh my god the reference to TJ """"Henry"""" Yoshi just killed me. I love you guys
me too. I can't believe the dude that makes educational videos about the deep inner workings of mario 64 was referenced by the guy that makes educational videos about the deep... hey wait maybe this isn't as big of a crossover as i though. i'm sure there's a pretty big overlap in audience
What reference is that? Also what video was it from? Was this from the one who talked about parallel universes in SM64 via a TAS? I forgot who I watched that did that
@@daniellewilson8527 yes that one
@@daniellewilson8527 the TH-camr you’re looking for is Pannenkoek2012
TJ “Henry” Yoshi getting dunked on once again
When I first saw the Dirac belt trick I thought it was flippant and didn’t explain anything. I still think so, but your video explanation was beautiful 😍 thanks for making this!
There's a video around showing a spin half particle has having an infinitude of belts coming from it to connect it to all points of the universe, making a continuous fibrous field instead of a scalar field. It's a nice visualisation about how the state of each point of the field includes the half-spin properties. I don't know if the electron is supposed to be spinning all the time and thus emitting ripples in that field, being electromagnetic waves including circular polarisation inline with the electron's spin vector.
@@tricky778 do you remember the title of the video? Sounds interesting
@@das_it_mane "electrons DO NOT spin" by PBS Spacetime. It didn't go into a lot of detail but it was interesting to see properties of their visualisation
@@das_it_mane it might have been "how electrons make matter possible" on the same channel
@@das_it_mane th-cam.com/video/eR9ZCwYPhhU/w-d-xo.html
So let's mark the electron with a '+'...
xD
was thinking the same thing
@@brooksbryant2478 charge is nothing but a ordinary element to distinguish between two different things we can say an electron is positive and a proton is negative it won't change anything
Arbitrary **
@@yourmom-nv9ui I know, it's just the opposite of the convention everyone else uses.
Not about charge.
This is a masterpiece. Thank you for making it. Please do more of it. Animations about the bloch sphere ans the Pauli matrices would be highly appreciated.
I felt very heavy vibes of the "turning a circle inside out" timeless video with the narration and imagery, especially at the first couple chapters.
Amazing work! As a layman i half half understood it, which is a gigantic feat!
not knot
that's a sharp corner
The color scheme of goldish-yellow and purple for the 2 sheets of the double cover seems like a direct reference to that video.
Well, TJ "Henry" Yoshi,
This video deserves to be seen over and over by anyone interested in the mathematical insight of spin. You are the first person to ever convey to me the right intuition for the Dirac belt trick. Keep up the great work, mate!
I think this may be the single greatest video on physics I have ever watched.
i came here for party trick and forgot why I came here, true masterpiece.
yeees another 3blue1brown style video, BRO we need more
When I first watched this video, I remember being very confused. After reading an introduction on Lie Groups, being reminded of this video and rewatching it, I get it now. Amazing quality.
you convinced me to watch the entire video with the watch for rolling rocks reference
This has such a 3B1B feeling to it... NICE WORK!
Big complement there. Agreed
Grant did the audio for the vid clearly.
56:50
now that's peak comedy
This is absolutely astonishing. Please keep making more mathematics/physics content like this. I have never seen these concepts explained so darn well!
Totally agree
OMG thank you for making a youtube that does NOT shy away from formula's, not even from high level math! There are so many youtubes about *interesting* concepts that in the end explain NOTHING because they restrict themselves to what they think every viewer should be able to understand (aka, nothing). This is unfair to the minority of people who CAN understand these concepts in terms of math (if explained well) and in general a disservice to humanity. That being said, after learning that my spacial insight got out of the graph and even off the edge of the paper by the professional trying to measure it; I have spent countless hours trying to imagine 4D space, thinking it has to be just lack of experience that humans "can't" imagine it (and because I suspected that it might be a reason that humanity is stuck with its understanding of physics where we are stuck). Imho it is more insightful to simply imagine a projection from cover spaces to its base space: picture the surface of a sphere as two discs at the same place: one being the projection of the upper half and one being the projection of the lower half, keeping in mind that each disc also has a "distance" (either up or down) to their respective part of the cover space. Putting the discs next to eachother is less insightful (although easier to show in a video). Obviously one then can only move from one disc to the other where this distance is zero: at the edge of the discs. Likewise and 100% equivalent: two spheres in the same place, connected at the edge (surface of the projected spheres) where the extra "up" or "down" distance is zero. Each point inside the two-sphere is then actually two points, where the distance (from the projected point in 3D space) is trivial: sqrt(1 - (distance to the center)^2).
yeah I agree. Even this could have done with way more, there was 0 discussion on what a group cover actually is or a rigorous way that shows SU(2)~S3
@@SuperMaDBrothers Well, if you want all the details, pick up an algebraic topology book and start reading. There's an optimal place to compromise, and this video pushed the boundary but earned the mathy bits with beautiful animations - without which, I'd much rather actually read a book.
Clicked on your profile pic because I thought it was funny but now I regret it. I gotta say, it's sad to see someone who clearly thinks so highly of their own intelligence and has somehow still fallen prey to blatant xenophobic propaganda. You clearly have the intellectual ability to figure things out, but lack the emotional maturity to see the reality for what it is. I hope you grow up and figure things out someday.
I know this is old, but regarding the part about the viewers knowing nothing: Yea many of us want to understand a concept and don't know much math (or are starting to learn). The reason why you're a minority is because the way these things are taught is backwards. If all of the science in schools today were approached with pure historic reference and first principles methods, then plenty more ppl would be able to understand it, and maybe even more would actually become scientists. Shying away from the hardcore math isn't great in some cases but it isn't bad either.
I keep going back to this video. It shows true pedagogical skill, showing that when the point that you are making is deep enough it deserves tender loving care even if that might seem to those who don´t care it might seem tedious. Go for clarity!
This is the best covering of this subject I’ve ever seen. Most of this material I’ve seen scattered across various courses like introductory topology (I had flashbacks when you put Hatcher on the screen), differential topology, non-relativistic quantum mechanics, or field theory, but no one’s ever put it all together like this with incredible visuals. Rotating an electron in the Black Lodge was just the cherry on top! I’m grateful for SoME 1 for putting this on my radar and I truly hope you do something like this again.
Thanks for mentioning the Hopf fibration.
This gonna blow up. This SHOULD blow up.
Thanks! You are illuminating the the first 300 or so pages of "math primer" in ""The Road To Reality" by Penrose.
I recognise that background music from another math animation channel ;)
The intuition that a belt is a path just flipped my world upside down... Only once, so it can never go back!
Wow just wow. I am bit older and my math degree is from the 70's. Damn I wish we had these beautiful visualization back then. I did a little bit with knot theory and would love to see this covered in a video.
This was really well paced, I had several moment of "oh that must mean ..." followed by the next section confirming it. Not had that experience in a while so it was an enjoyable journey.
'A large book'
*Serge Lang's 'Algebra' emerges*
The haircut in the beginning is a double cover of the one in the end :). Great video!
3:42 this is the best form of comedy there is. very well done
also great video btw
There's another great one right at the end, too.
Putting the electron in _that_ room was very chef's-kiss, as they say.
I think minutes 0-18:48 should be mandatory viewing on day 1 of a topology class. Starts with a physical phenomenon that's cool; builds up a space that motivates ideas of quotients of topological spaces and manifolds (and identifying antipodal points of spheres ---> real projective plane, Boy's surface, etc.!), then studies loops on that manifold motivating ideas of homotopy theory like contractibility (relative to some fixed endpoints) visualized in different manners, all still grounded in the hands-on real world by Dirac's belt trick.
And of course all the topological content afterward: spheres as 2 disks of the same dimension glued along the boundary (i.e. forming the sphere as an adjunction space/categorical pushout of a diagram), covering spaces, the lifting lemma... truly a wealth of content here, all presented in a welcoming way! Even your proof by contradiction was presented in a welcoming way; I'm pleasantly surprised that one can make rigorous arguments with just a 40 minute "picture based" introduction to topology.
The summaries were also really nice! Lecturers don't do those often enough in classes, I think.
IMO they should first introduce Topology with this sort of clearer material and call it 'Apology'
thanks for content bruh
Wow! That was so astonishingly beautiful... the kind of quality I have come to expect from 3Blue1Brown... While I have an MSc in Computing Science, I was actually pretty good at math and physics as an undergrad, and continue to try to better understand quantum physics. Spin is so hard to wrap my head around (pun intended), but this really gave me such a good feel for what might be going on, a glimpse in the nature of quantum mechanics. By the end of your video, I could really appreciate how particles have angular momentum, and why fermions are so special. Thank you so much for opening my eyes...
If you try to understand the spin of an electron by looking at it from all angles, you won't get it ;).
One of the best movies I have ever seen on this complicated topic. Noah you are a true genius. Keep them coing
The probability of finding such a video on youtube is 1 in a billion :-) amazing ..
This is definetly the best video to explain SU(2) and spin at a fundamental level of all YT
Had to take several days to watch this due to time but, the realization how everything he explains relates to the belt and quantum mechanics around 45:00 felt like a hit of heroine. The satisfaction of this just completely washed over me
i love that you put the electron in the black lodge
I feel so stupid when I listen to your lecture...I understand just nothing, but it stimulates my old brain and teaches me humility. Thanx
This is an amazingly well produced and didactically superbly laid out video! Kudos!
ohh, 3blue1brown style, I love it! You are doing a great job in using manim
Came from 3b1b's competition. Great video!👍🏾
Hi Noah, your videos and notes are really awesome! i read your A Crash Course in Statistical Mechanics and couldn't believe how useful it is for me as a high school student doing my independent research project in computational chemistry! please don't stop making those notes and videos you are so talented in explaining complex physics intuitively!!!
Truly amazing video! Really enjoyed the philosophic conclusion.
Thank you. I'm a senior maths student and just learned about group theory and have always been confused when I heard SU(2) and SO(3), thank you for this intuitive explanation!!
Algebraic Topology applied to Quantum Physics: automatic subscription for me. Keep it up!
i love to listen to ans watch your explanation when I'm sleepy and even more when I'm fully awake.
Once again, one seen this explained in an over simplified way so many times that leaves so much out. I think this is simplified as much as it can be while still giving some useful insights to a non-expert. Thank you.
Phenomenal video. I'd disagree at 54:41, yes the phase of one part of the electron superposition matters, but that is a local phase. A global phase (if that's what 'overall' means), would affect both parts of the superposition, and is undetectable. A similar example is putting glass on one slit of a (laser) double slit experiment, that shifts one path's phase and moves the interference pattern. Glass on both slits shift both phases equally and doesn't change the interference pattern at the (distant) screen.
I agree with you completely. If we are tensoring the spin Hilbert space with the center-of-mass position Hilbert space then a truly global phase could not affect any measurement. Really there would be a relative phase between the basis states our full state is a linear combination of. Having said that I couldn't figure out a good way to say this in a way appropriate for the level of the video and opted to say something which was evocative of the truth yet slightly wrong, with the expectation that people "in the know" wouldn't be thrown off
@@noahexplainsphysics THE MATHEMATICAL PROOF THAT ELECTROMAGNETISM/ENERGY IS GRAVITY (IN WHAT CONSTITUTES A BALANCED FASHION) IS SUCCESSFULLY (AND CLEARLY) DEMONSTRATED, AS E=MC2 IS F=MA:
The Maria ("lunar seas") occupy ONE THIRD of the near side of the Moon. The land surface area of the Earth is 29 percent, AND this is EXACTLY between one quarter AND one third. The Moon is about one quarter (27 percent) the size of the Earth in what is a predictable fashion, AS it is FULLY manifest as LAND. The sky is BLUE, AND the Earth is ALSO BLUE. So, consider BALANCED BODILY/VISUAL EXPERIENCE. LOOK at what is the orange Sun, and think LAVA. Gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Accordingly, the rotation of WHAT IS THE MOON matches it's revolution. Now, consider the fully illuminated (and setting) Moon in direct comparison with the orange Sun. They are both the size of THE EYE. Notice that the Moon is ALSO BLUE. Consider the man who IS standing on what is THE EARTH/ground. Touch AND feeling BLEND, AS ELECTROMAGNETISM/energy is gravity; AS E=MC2 IS F=ma. The BULK DENSITY of the Moon is comparable to that of (volcanic) basaltic LAVAS on the Earth. The energy density of LAVA IS about three times that of water. Great. The human body is, in fact, about as dense as WATER !!! Think !!!
"Mass"/ENERGY IS GRAVITY. ELECTROMAGNETISM/ENERGY IS GRAVITY. THE SUN AND THE EARTH are F=ma AND E=MC2, AS ELECTROMAGNETISM/ENERGY IS GRAVITY. E=MC2 is DIRECTLY AND FUNDAMENTALLY DERIVED FROM F=ma. F=ma AND E=MC2 PROVE that ELECTROMAGNETISM/ENERGY IS GRAVITY, AS ALL of SPACE is NECESSARILY ELECTROMAGNETIC/GRAVITATIONAL (IN BALANCE); AS ELECTROMAGNETISM/ENERGY IS GRAVITY. "Mass"/ENERGY IS GRAVITY. ELECTROMAGNETISM/ENERGY IS GRAVITY. Energy has/involves GRAVITY, AND ENERGY has/involves inertia/INERTIAL RESISTANCE. Gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE, AS ELECTROMAGNETISM/ENERGY IS GRAVITY. This explains F=ma AND E=MC2, AS ELECTROMAGNETISM/ENERGY IS GRAVITY. SO, GRAVITATIONAL FORCE/ENERGY IS proportional to (or BALANCED with/as) inertia/INERTIAL RESISTANCE; AS ELECTROMAGNETISM/ENERGY IS GRAVITY. ACCORDINGLY, ALL of SPACE is NECESSARILY ELECTROMAGNETIC/GRAVITATIONAL (IN BALANCE); AS ELECTROMAGNETISM/ENERGY IS GRAVITY. "Mass"/ENERGY involves BALANCED inertia/INERTIAL RESISTANCE consistent WITH/AS what is BALANCED ELECTROMAGNETIC/GRAVITATIONAL FORCE/ENERGY, AS ELECTROMAGNETISM/ENERGY IS GRAVITY. GREAT !!! Gravity IS ELECTROMAGNETISM/ENERGY. ELECTROMAGNETISM/ENERGY IS GRAVITY. INDEED, A PHOTON may be placed at the center of THE SUN (as A POINT, of course); AS the reduction of SPACE is offset by (or BALANCED with) the SPEED OF LIGHT (c); AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/ENERGY IS GRAVITY. GREAT. "Mass"/ENERGY IS GRAVITY. ELECTROMAGNETISM/ENERGY IS GRAVITY. ALL of SPACE is NECESSARILY ELECTROMAGNETIC/GRAVITATIONAL (IN BALANCE), AS ELECTROMAGNETISM/ENERGY IS GRAVITY; AS E=MC2 IS F=ma. Great. The ability of thought to DESCRIBE OR RECONFIGURE sensory experience is ULTIMATELY dependent upon the extent to which THOUGHT IS SIMILAR TO sensory experience. (THOUGHTS ARE INVISIBLE.) Gravity IS ELECTROMAGNETISM/ENERGY. ELECTROMAGNETISM/ENERGY IS GRAVITY.
Very importantly, outer "space" involves full inertia; AND it is FULLY INVISIBLE AND black. The perpetual motion of WHAT IS THE EARTH is NOW explained. GREAT !!! The idea that THE PLANETS are "falling" in what is "curved space" in RELATION to what is THE SUN is PROVEN to be NONSENSE. So, the falling objects must be considered in RELATION to WHAT IS THEN THE ORBITING EARTH. GREAT !!! E=MC2 IS F=ma. The stars AND PLANETS are POINTS in the night sky. I have explained why the motion of WHAT IS THE MOON matches it's revolution. E=MC2 IS F=ma. This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE, AS ELECTROMAGNETISM/energy is gravity.
Consider the man who is standing on what is THE EARTH/ground. Touch AND feeling BLEND, AS ELECTROMAGNETISM/energy is gravity; AS E=MC2 IS F=ma. Gravity IS ELECTROMAGNETISM/energy. Gravity AND ELECTROMAGNETISM/energy are linked AND BALANCED opposites, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. GRAVITATIONAL force/ENERGY IS proportional to (or BALANCED with/as) inertia/INERTIAL RESISTANCE, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE, AS E=MC2 IS CLEARLY F=ma; AS ELECTROMAGNETISM/energy is gravity. Accordingly, the rotation of WHAT IS THE MOON matches it's revolution. Great. (Gravity IS ELECTROMAGNETISM/energy.) E=MC2 IS F=ma ON BALANCE. "Mass"/ENERGY involves BALANCED inertia/INERTIAL RESISTANCE consistent WITH/AS what is balanced electromagnetic/gravitational force/ENERGY, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. So, objects AND MEN fall at the SAME RATE (neglecting air resistance, of course); AS ELECTROMAGNETISM/energy is gravity; AS E=MC2 IS F=ma. A given PLANET (INCLUDING WHAT IS THE EARTH) sweeps out EQUAL AREAS in equal times consistent WITH/AS E=MC2, F=ma, AND what is PERPETUAL MOTION, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy. The stars AND PLANETS are POINTS in the night sky. Excellent !!! E=MC2 IS F=ma ON BALANCE !!!
TIME dilation ULTIMATELY proves ON BALANCE that E=MC2 IS F=ma, AS ELECTROMAGNETISM/energy is gravity. INDEED, TIME is NECESSARILY possible/potential AND actual IN BALANCE; AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. It all CLEARLY makes perfect sense, AS BALANCE AND completeness go hand in hand. Gravity IS ELECTROMAGNETISM/energy. E=MC2 IS CLEARLY F=ma ON BALANCE !!! Great !!!
By Frank DiMeglio
@@frankdimeglio8216 I can feel my brain melting
Actually really impresive. Bravo!
This was really helpful. Looking forward to more if you have the time.
That was amazing and my no 1 of the SOME so far! Thanks a lot for taking the time to explain it!
This is an absolutely brilliant video! I am so glad to find this channel thanks to the SoME1.
Absolutely fantastic video! Genius presentation - thanks so much for putting this together!
An absolutely delightful video, congratulations!
Thank you for making this! I was trying to wrap my head around the whole so3 and su2 thing and was just searching youtube for any visualization. Didnt expect to find something so high quality!
Totally amazing video. Best video I have watched in months.
Bravo! You made an AMAZING work!
Great explanation! Thank you for making the video!
this was a great vid thank you for your efforts in posting.
Wonderful video. Your animations and script are very methodical without being boring. Your video reminds me that, as David Hilbert once said, 'A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street'. Well done.
Better than watching a movie 🍿Thanks for the video :D
Thank you for explaining the basics and the notation, I really needed that.
Wow, this is a masterpiece 😍 I especially like the last part on the Hopf fibration
It's so cool how you've used manim creatively here, especially with the white theme instead of the dark one
This is a phenomenal video, it's so intuitive
This is really well done, thanks for uploading it.
Please make more videos like this! It feels like 3Blue1Brown but for physics, which I'm sure for many people is even more interesting. The video was amazing and interesting, so thank you.
Try minute physics
The connection I am making here is that the 3-sphere being a double cover of rotations is why the unit quaternions (a 3-sphere) are so good at rotation (SO(3)).
Yes, I think that's right. They give a much more convenient parameterization of rotations at the cost of having an arbitrary sign
Nice work - I loved it! Great use of animations. I can't believe how similar our videos are.
with this video, you just stepped up your game Noah!
Thank you sir, keep continuing such amazing videos and interesting subjects. Please before the spotlight hits you and your channel (which I think is just a matter of time) always strive for quality and thoroughness of your videos over anything else.
thanks so much man I'm at 12 min and the video up till now is so much insight. I wanted to investigate paths in spaces of rotations/lie groups as well as quotient topologies for a while but I've been distracted from it, thanks again!
Amazing video, super interesting, keep going!
I can't express how absolutely taken away I am by this video! Fantastic animation, amazing narrative! I had that feeling of awesome math discovery throughout the whole video, thank you so much for putting in an immense amount of effort and love into this video!
I was concerned when he said “for professionals” and he still explained everything he did beautifully
That was excellent. Thank you.
Literally the best math video I have ever watched. Thank you, so much.
Wow, alot of info packed in here! Well done!
One thing I'd like to mention is that the figure on the cover of Hatcher's Algebraic Topology is the Hopf fibration.
Also, many find Hatcher's to not be rigorous enough. There are plenty of more formal treatments, but Hatcher has better examples and exercises than all other AT books combined. To me that's worth it. But to each their own, I guess.
This was amazing, thank you!
This is excellent. Well structured and challenging. Understandably this took great effort to put together but I hope there will be more like this.
This is insane video! You've explained it at the right pace and splendidly! I'm surely gonna recommend your video through all my friends
I have no words to explain how good is this.
I mean, I am reading The Road to Reality, specifically chapter 15 which is dealing with these matters. My background is telecommunications, so group theory is a bit alien to me. How helpful is this, I think Sir Roger Penrose would be utterly pleased by this video. I am sooooo curious what he would say.
Thanks a quintillion!
Amazing work
Love the explanation and animation!
This is GREAT, I think I'll go back to watching it every once in a while. It is brilliant. Thank you
Amazingly explained, thanks a lot! But I have to say I watched it twice, because the first time I left me with a negative impression ;) Keep doing videos like this one please!
is there any chance of u making videos on.....drum rolls.................. literally every topic of physics. I'd literally love ur videos on (statistics and physics) .
Exceptional explanation! A+ for you sir.
Very nice illustration crystal clear.
Fan-ta-stic! Thanks a lot for this amazing video. As a quantum mechanics teacher, I will strongly recommend it to my students and... my collegues too! This a really great job. Many thanks again :-)
Great work. A worthy description of the content of the video can also be seen in John Baez's "Gauge, Knots and Gravity", or in the more brief lecture notes on spin.
Appreciate the background music 👍
unparalleled explanation skills, suited for an actually high level audience!