Dirac's belt trick, Topology, and Spin ½ particles 

Fun fact: All USB ports before USB-C have spin 1/2

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.

"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

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!

Oh my god the reference to TJ """"Henry"""" Yoshi just killed me. I love you guys

TJ “Henry” Yoshi getting dunked on once again

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.

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!

yeees another 3blue1brown style video, BRO we need more

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!

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).

Thanks for mentioning the Hopf fibration.

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.

"Wait, a 360-degree rotation is a 360-degree rotation, you can't say it's only a half!" "Well, Albert """Henry""" Einstein..."

I think this may be the single greatest video on physics I have ever watched.

This is absolutely astonishing. Please keep making more mathematics/physics content like this. I have never seen these concepts explained so darn well!

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.

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!

Thanks! You are illuminating the the first 300 or so pages of "math primer" in ""The Road To Reality" by Penrose.

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.

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 has such a 3B1B feeling to it... NICE WORK!

you convinced me to watch the entire video with the watch for rolling rocks reference

'A large book' *Serge Lang's 'Algebra' emerges*

This gonna blow up. This SHOULD blow up.

One of the best movies I have ever seen on this complicated topic. Noah you are a true genius. Keep them coing

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...

This is definetly the best video to explain SU(2) and spin at a fundamental level of all YT

The haircut in the beginning is a double cover of the one in the end :). Great video!

So let's mark the electron with a '+'... xD

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.

The intuition that a belt is a path just flipped my world upside down... Only once, so it can never go back!

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!

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!!

ohh, 3blue1brown style, I love it! You are doing a great job in using manim

I recognise that background music from another math animation channel ;)

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.

It's so cool how you've used manim creatively here, especially with the white theme instead of the dark one

The probability of finding such a video on youtube is 1 in a billion :-) amazing ..

This is an absolutely brilliant video! I am so glad to find this channel thanks to the SoME1.

Well, TJ "Henry" Yoshi,

You belong to the Group of "Great Explainers".Thank you very much for a very clear explanation of a rather abstract concept.The best I have seen sofar.

Putting the electron in _that_ room was very chef's-kiss, as they say.

3:42 this is the best form of comedy there is. very well done also great video btw

IMO they should first introduce Topology with this sort of clearer material and call it 'Apology' thanks for content bruh

unparalleled explanation skills, suited for an actually high level audience!

legendary video, my man summarizes at the end of each section, great technique bless you bro

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!

Very good video. Clear and motivacional. It is not an easy topic to explain for those with no basics on algebraic topology, but quite illustrating. Congratulations.

with this video, you just stepped up your game Noah!

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!!!

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 :-)

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.

Came from 3b1b's competition. Great video!👍🏾

Algebraic Topology applied to Quantum Physics: automatic subscription for me. Keep it up!

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!

i came here for party trick and forgot why I came here, true masterpiece.

You are an outstanding teacher! Part 2 is by far the best visual illustration of the 4 pi concept. I know that required a large amount of work on your part and your viewers thank you. If I understand correctly, there in part 6 there is a need also to introduce the sphere S^3 as a disjoint sum of upper and lower disks D^3 mod equivalence of common boundary; S^2 = shared boundary of upper and lower D^3 disks.

This is the most fantastic video I've ever seen on youtube. I mean fantastic in the positive sense. It is absolutely mind-blowing. But the greatest miracle of all is that: It is understandable. Even someone like me, who struggles alone for years with these concepts, could follow everything in it.

Totally amazing video. Best video I have watched in months.

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

This video is a wonderful symphony comparable to all cosmic music Thank you from the heart, thank you very much

i love to listen to ans watch your explanation when I'm sleepy and even more when I'm fully awake.

56:50 now that's peak comedy

This video should be getting a lot more views

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.

Thanks Bro. For the past 3 months I was struggling to understand what quantum spin is in terms of Topology and Group Theory. Thanks for connecting the dots with a clear explanation. Great teaching.

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 amazingly well produced and didactically superbly laid out video! Kudos!

i love that you put the electron in the black lodge

Ooooh, this looks good. I've been looking for a more thorough examination of symmetry groups. I'm bookmarking for later viewing.

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!

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

Absolute amazing. Last time I learned about spin was... 7 years ago, and this not only refreshed, but solidified my understanding of spin 1/2.

Awesome video! For section 2, it might help make clearer what you're saying/doing if you point out that you can translate and scale the belt any way you like. Demonstrate that, and it becomes clearer why a single twist can't shrink to the origin: because it cuts through the edge of the sphere one time, and can't "undo" that. A two-twist belt cuts through the edge twice, forming a loop. Distort that loop as you do in the video, to show it as a loop that cuts the edge in two places, then translate to remove the cuts, then shrink to the origin.

Wow, this is a masterpiece 😍 I especially like the last part on the Hopf fibration

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!

Tricks beautifully explained, Maestro!👍👍👍👍👍

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.

I was concerned when he said “for professionals” and he still explained everything he did beautifully

I wish you will be one of the contest winners! Exceptional excellent mathematical physics video!! Bravo!!

Whoah you're a god-tier, please make more :O

Can't comment, We need to implement this education level here in India. This videos will suerly helpful

That was excellent. Thank you.

Without doubt, this is one of the best (if not THE best) video on this topic that I've ever seen. A big thank you!

Better than watching a movie 🍿Thanks for the video :D

Truly amazing video! Really enjoyed the philosophic conclusion.

How on earth does this video have this many likes? It’s AMAZING and you should really keep up with these videos, in the same wonderful way you are doing now ♥️

WoW space of all rotation in a sphere with radius pi. This is a great eye opener. Thanks!

Literally the best math video I have ever watched. Thank you, so much.

Your video is of outstanding quality. Maybe a bit advanced for a general audience, making it hard to appreciate if you are not a physicist like myself. Keep making videos like this !

Never realised that SO(3) is the three dimensional projective space... Neat!

I LOVE the nineteen fifties style memory effects at 13 minutes? Great video!

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

Noah , this was truly excellent. Thank you for making it and doing the faq and video indexing too This brought together in a single presentation many of the concepts I’ve encountered over the years creating a type of map or perhaps a trail of breadcrumbs to be followed. I still don’t understand the nature of electron spin, but you’ve provided a wonderful foundation for appreciating the mathematics not usually discussed when looking at Dirac’s solution of relativistic wave equation. ( Dirac , Principles of QM, 3rd Ed Chap XI ). As with any good map, to appreciate one must make the journey. I’m sure I’ll be looking at this video many times as I do that. Best EC D. ps : I see you have another video on this topic. Thanks in advance.

Struggling in trig right now and this helped me with the massive dose of perspective I needed to make it click

3:42 might be the best reference I've seen in a while. Also, nice video

Best video I've seen on this topic so far. The tying up loose ends was right on the nose regarding questions I had!