I know how hard it is to work with manim, assuming that’s at least partly what you used. And to explain one of the hardest topics ever with it? To the public? With a 70-minute video? ~3 months is honestly an impressive duration. I am sure it’s a great video too, I haven’t finished it yet but that’s my first thing to do tomorrow. Thank you and keep up the good work 😁
@@RichBehiel Topological holes cannot be shrunk down to zero -- non null homotopic (duality). Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Inclusion is dual to exclusion -- the Pauli exclusion principle is dual. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Antipodal points identify for the rotation group S0(3) -- Dual perspectives! Points are dual to lines -- the principle of duality in geometry. "Always two there are" -- Yoda. Duality creates reality! Spinors are mobius loops. The Klein bottle is composed of two mobius loops -- self intersection. The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle. Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates.
This to me is another stellar example of why youtube may be one of the greatest libraries of human knowledge ever collected in a single place. Congrats on offering up your tome, and for supporting the spread of quality information.
I recently got a PhD in atomic physics and found this extremely enlightening on concepts that I took for granted all this time. I wish videos like this existed when I was in grad school.
@@redflipper992 , I leaned back and thought to myself: "The entity that traveled almost 2 years back in time to deliver me a message about the racism and Human Violations I was going to receive. And why would they travel that far , in salutation for [me] , only to allow kkk , clandestine, Jim crow, etc. to continue stealing, like, for instance, brain waves (why did I allow my Nigerian doctor to hook me up like that?) and other devices utilized for evil?" Then it came to me. These entities are watching too, waiting to kill these wicked thieves something terrible and unimaginable. I felt much, much better.
I wondered from the first one I ever saw, “how could you start with a DB-9, and get to THIS, and call it better?” Guess they didn’t employ any mechanical engineers…
Wow. This is what youtube should have been. Not youtube shorts that rot my brain chaining me to scroll endlessly for miniscule amounts of dopamine and serotonin. Thank you. Honestly. Thank you.
You can get it to stop showing you shorts all the time if you tap the three dots on all the ones that show up on your homepage and tap "not interested". It'll eventually start recommending them again, but less and less often the more you tell it "not interested" in every short it puts on your homepage.
I know this was for a primarily physics audience, but I have never had SO(3), SU(2), quaternions, and spinors explained to me so clearly in any video ever. As someone from more of a programming background with interest in rotations and vectors from an algorithmic perspective, I've vaguely known about quaternions and matrices and their relation to rotation. But never have I ever had these objects explained in a way that I well and truly understood in a way that I could explain to others. I still am not 100% on the link between quaternions and spinors since you kind of glossed over it here, but I feel like I've definitely taken a major step in being able to get it. The mathematicians out there should learn that rigor is not explanation! I've seen videos that rigorously explain what spinors are, precisely, and I kind of got it. But I never made the connections on how all the parts really fit together until this video. So thank you! For me, it's all about understanding the motivations and framing the concepts in a way that you "discover" them on your own. That's how you build true understanding. You did an amazing job of that here.
"If you get this concept about the two homotopy classes, if you really feel it, then instinctively you'll suspect that maybe there might be some mathematical object that is sensitive to the homotopy class of rotations... you'll yearn for it" I can tell you've done an incredible job of setting up the intuition for this subject because that was exactly what I was thinking by this time in the video.
What an amazing setup that was. By the time he named the weird little 4d vector thingy as the spinor, I shouted 'hah' out loud, and my wife gave me the look. Totally worth it!
I wasn’t expecting such a deep/philosophical dive at the end with the spin-statistics theorem. I left inspired after watching the whole video. Appreciate such masterpiece.
Maybe the most interesting youtube video I've ever watched. As a layman I've always had a casual interest in these topics, this tied together so many things I was curious about but never quite grasped. Thank you for your work!
@@ExpandDong420 No. It is things you can squeeze together to a ball, and things you cannot squeeze together to a ball. Anything that has a hole in it, cannot be squeezed into a perfect ball, the hole will still be there, even if it is smaller, a rift or whatnot. And so comes the conclusion, there are two shapes, and every item is one or the other. And so the question: Is the universe one or the other, which shape is the universe. In formality, a priori, knowing nothing about the universe, it's a coinflip. It could just as well be one as the other. The chances the universe is like a teacup (with a handle, that makes a hole in the shape, so you cannot squeeze it into a perfect ball), is 50%, and the chances that it is like the coin that you flip, that can be squeezed into a ball, is also 50%. The shape with the handle, is what gives birth to the idea of a wormhole. Which we can then say, has a 50% chance to be a possibility, in our universe, a priori, knowing nothing about the universe. It really goes to show, how farfecthed and thinly veiled in philosophy, the idea of a wormhole is, given that it has become a thing in the imagination, put beside established fact in the minds of man. If the universe has a handle like a teacup, can you then travel along that handle, to show up somewhere completely different than where you started?
This is SO. MUCH. WORK. How did you get this video out the door? My god the animations!! Here's hoping that if there is a day job in your life, it is paying really well. This is waaaaaaaaaay more valuable education than I can get from paid sites.
Thanks! Yeah it was a lot of work 😅 But the ratio of time spent working on it, to the time that people will spend watching it, is a really good deal! It’s a great way to put positive vibes out into the world.
@Shplinkinshploinkin good point! That animation was by Jason Hise. He published it copyright free, so anyone can use it. I put his name in the thumbnail and later on in the video when that animation comes up. Credit where credit is due! :) I rarely borrow from others, but his animation was too beautiful not to include. I did all the other animations though.
Again, the most pellucid explanation on the topic you cover. Last time complex numbers, and the Dirac equation. This time, spinors. Bravo, Richard. Bravo.
Correct use of the word pellucid gets a bravo from me! I just gave myself a headache distinguishing perspicacity from perspicuity, all the while forgetting that pellucid is euphonically lovelier than either!
The fact that you provide this wonderful masterpiece of a video for free for everyone to see and study says a lot about your character. Thank you so much for your dedication to science and education!
Here is some intuition on the two rotation types hopefully: Grab some object ( not needed, you can use an empty hand, but an object makes visualisation way easier ) Without re-gripping the object, If you make a class II rotation, the object will end up in the same orientation ( by definition ), and your hand can also end up in it's starting orientation. If you make a class I rotation, the object will end up in the same orientation, but your hand will end up twisted, and the only way to fix that is to make a second class I rotation ( or re-grip the object ). In all cases, you can just make the exact same class I rotation again. ( may not be obvious at first how to do that though ) The two class I rotations together form a class II rotation, which means that your can end up how it started If you do that a few times with different rotations, there is a 90% chance that you can now intuitively differentiate which rotations are class I and which are class II just by looking at them. You have also just demonstrated having to turn an object ( your hand ) around twice for it to end up in its original state. This happens because it is connected ( with specific constraints ) to an object which itself cannot rotate.
Ok I’m playin with this but it doesn’t make total sense. It seems like class II rotations I have to move my entire body around the object, like walk in a circle, whereas class I rotations my body stays still and my arm rotates (rather uncomfortably) and then I have to regrip to get my arm and hand back to its original configuration. Is this correct?
I was just looking to learn more about quaternion and now I have some existential crisis over the fact that their "square root" hold the universe together by preventing some atomic collapse. Great job
Finally! Someone explains spinors in a way I can understand! People have been trying to tell me stuff like "Spinors are like the square root of Geometry." or "Spinors are these special vectors that satisfy a certain requirement.", or other statements that mean nothing to me. But you. You just simply said "Spinors are vectors with complex components." and it all made perfect sense. I actually think about complex numbers and quaternions all the time, so explaining stuff in terms of complex numbers and quaternions is quite useful for me.
Thanks for the kind comment, and I’m glad you loved the video! Hopefully I can be up there with those guys someday. I’ve got some catching up to do, though! Working on the next video now :)
Topological holes cannot be shrunk down to zero -- non null homotopic (duality). Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Inclusion is dual to exclusion -- the Pauli exclusion principle is dual. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Antipodal points identify for the rotation group S0(3) -- Dual perspectives! Points are dual to lines -- the principle of duality in geometry. "Always two there are" -- Yoda. Duality creates reality! Spinors are mobius loops. The Klein bottle is composed of two mobius loops -- self intersection. The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle. Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates.
@@hyperduality2838 Thank you 🙏🏻 for the comment. It was stimulating. I liked the parts about spinors, Mobius loops, and Klein bottles from topology. I have thought about these things as definitely related, I didn’t think anyone else had. 😂🤷♀️
Another tip of the hat to your brilliant pedagogy. Without you even trying the first few minutes of your video will surely be the ah ha moment for those struggling to grasp simplicial complexes, vietoris rips and persistent homology. It’s better than anything I’ve seen on UA-cam actually trying to “explain” the subject! Let alone the rest of the video. Just WOW
This is so dope. Like, easily one of the most impressive videos I've seen on youtube. It's elegantly concise, with relatively minimal assumptions about prior knowledge. Plus the whole thing's informed by an simply communicated, profound intuition for the originary principles of topology. Amazing way to frame an introduction to spinors. Not to mention so many amazing labor intensive animations. Awesome aesthetic intuition throughout. Rare to see so much work go into a video that truly privileges' a generous pedagogy. Been a huge help for me today in creating tangible mental images as I pursue self-education in an intimidating subject. Sure thousands of other folks will feel the same gratitude. To me, nothing's cooler than someone who works so hard to share their hard earned skills like this. Especially when it's given to everyone for free. Thanks dude!
I was thinking of how does something move from one Planck length to the next because of the discreet nature of it. Your presentation on Spinors here gives me some food for thought! I'm super amateur, but I love thinking about this stuff! I can't wait to finish this video after work and get to the part about the complex numbers. There may be a connection to a thought I was having about higher dimensions and branes. Thank you very much!
I don't know enough to comment on the quality of the content, but this is amazingly presented and VERY interesting. Thank you for the obviously large amount of effort you put into conveying this abstract necessity.
What a good video. I especially love the audible frustration at around 34:00 about *what, why would you need anything beyond this*, it's a very relatable feeling.
Bravo. Well done. I was reminded of a mysterious native-Colombian practice that MAY be experimental evidence of applied spinor theory. What comes to mind is my recent experience meeting a delegation of Kogi (of Santa Marta, Colombia) a surviving native culture with generational teachings based on a deep connection to nature over millennia. My understanding, is that the Kogi rarely travel, but do so as a mission to bring balance to the unbalanced world. The Kogi taught the guests a practice of communication with nature that immediately resonated with my modest understanding of spinors. Specifically, when the Kogi communicate with nature (e.g. a tree), with a balanced and open mind, they keep directed ( to and from the tree) intention in mind and spin twice to transact the balanced communication. On learning this practice, I was immediately reminded of spinors. I am still resolving my feelings regarding this practice and a correspondence between intention (as a real experience described by an exclusive thought pattern) and fermions (as a real particle described by the Pauli-Exclusion principle). My musings have lead me to visualize the local (human) intention and remote (tree) "intention" as two fermions with opposing spins that temporarily entangle (perhaps as a low resistance superconducting Bose-Einstein-esq spin ice?) to transact the communication. My guess, is the (personal) experiment may be evidenced by noticing a reduction (maybe by half) of original human-intention as a feeling and an increase in an awareness of "tree-intention". Anyway, this is a work in progress and I felt a need to share... Thank you for your video and making contact with the mysteries involved.
This is a wonderful comment. I'm instantly reminded of a video where they were talking about the Native children and how they were more equipped to understand Einsteins theory thru their experience with nature. Have you seen the Fermionic Mind Hypothesis? There's definitely something here to your idea. I like this sort of out of the box connection drawing.
You have just given me the extremely rare and thus appreciated sensation of: "Yeah, I know where this is going, I have dealt with this befo... OOOoooooh, that's a neat way of looking at it!"
3blue1brown has been real quiet since this dropped. The animations are astonishing and I'm impressed by how efficiently and continuously you were able to explain this difficult idea. Good job
this is an incredible video. i'm a computational chemist that regularly does quantum mechanical calculations but i've never had a chance to really peek under the hood of the code i run. this is a great starting point!
Engineer here, run for cover. It seems that the spaghetti animation is showing something interesting that I'm not sure has been made explicit here or elsewhere. The 720 rotation has two phases where the first phase has the spaghetti strand is going over the origin and the second its going under the origin, and that might visually explain the double cover. I've seen that type of animation a good number of times, but haven't noticed that connection until watching this video, and haven't yet heard it explained in that way. No doubt this may be obvious to those familiar and comfortable with the ideas. Thanks for the video, good stuff.
These visual representations that you have created for this video are incredibly good. I'm a computer scientist and not a physicist and yet I feel that you've made it easy for me to follow along with the concepts.
I finished my master's in physics eng. about a year ago and sadly I don't think I will come back to academia after that. But this video is so inspiring it makes me want to be a student again
I love the way this is simplified by the host!! The comedic relief is a blessing as well. And the mystery of the spinors is my favorite part!! I think spinors could easily explain the mental and astral planes in relation to the physical plane. But first I need people to understand these formulas. lol.
This is amazing. Pure youtube gold. I remember studying group theory two years ago and being confused by the 2 homotopy classes of SO(3) and the fact that SU(2) was it's double cover. Now, thanks to you, I can say that I visually understand that, but the simple confusion evolved into an existential confusion. By the way, loved the bits of humor included every now and then in the video.
Regarding 44:52. Gimbal Lock is not an issue inherent to SO(3), and using SU(2) does not avoid it. That's a common misunderstanding that gets repeated a lot. Gimbal Lock can happen when you naively attempt to compose a rotation using Euler Angles. It's an issue with the *mapping* from Euler Angles to the space of rotations. It doesn't matter if you use SU(2) or SO(3); if you try to compose a rotation from Euler Angles, you can run into Gimbal Lock.
Interesting! Thanks for the correction! I’m confused though. Won’t any representation of SO(3) have poles, which would present a problem for a 3-gyroscope system? Whether using Euler angles or axis-angle vector. Or are you saying that, suppose we had a four-gyroscope system, we could still use SO(3)? I see how we could do that in a roundabout way through SU(2), but I’m struggling to see how to do it directly in SO(3) without the poles being a problem.
@@RichBehiel Tait Bryan angles, AKA "roll pitch yaw" used in aircraft attitude. Since I've written tons of code for rotations in quantum mechanics, remote sensing and mars landings...I've used many reps (and spacecraft GNC ppl use "quaternions" and if you said SU(2) repression they would have no clue what you're talking about). Anyway, any rep that has 3 different axes shouldn't be degenerate, and requires a computer to invert. I am convinced the only reason Euler (an absolute genius, ofc) used repeated axises, thus allowing degeneracy, is that his iPhone had a lousy chip and he wanted to be able to invert rotations, which for Euler angles is (alpha, beta, gamma) --> (-gamma, -beta, -alpha)..no chip required. In a related note, doing geodesy, I also learned we have the first, second, and even 3rd eccentricity, flattening, etc..is because early cartographers couldn't do definite elliptic integrals with parchment and an inked up feather.
@@RichBehielI would see the Stack Overflow post titled "Quaternion reaching gimbal lock". The first and second answers are very elucidating. The way a rotation matrix stores a rotation is not via axis angle or Euler angles but by storing the orthonormal basis that results from the rotation.
I have only an elementary understanding of math and physics but nonetheless I watched this richly complicated and masterfully created video to the end because at a very early moment in it I was struck by the intuition that herein lies something of great value and immense importance to understanding our reality. The quote from the Spin Statistics book at the end confirmed my hunch and I am glad that I persevered to the end though that is not to say that the insights gained along the way were insignificant. Thanks to Rich for putting what to us who are toddlers in his world seems like several lifetimes of unimaginable hard work into this masterpiece of exploration and explanation into the mysterious nature of spinors. His humble and humorous way of presenting his work is very endearing and makes for us babies in the physics world what would otherwise be indigestible content something that we want to roll around in our mouths and suck upon even though we can never hope to bite down on it and really get to the marrow of the matter. So thank you, thank you!
Excellent presentation! I have a phd in theoretical physics too. It was a long time ago and I am rusty, so I have reverted to be only a little past the denial stage :)
I get it! I finally get it! All the talk of a belt, or the Balinese candle dance, or a little cube inside a big cube and connected by rubber bands, I could never visualize. But I can see that a wiggle is the same as an octopus!
The "belt/plate trick" isn't a trick. It's a fair intuitive/visual account of spinors. A spinor isn't a rotation of a free object. It's a rotation of a tethered object, as illustrated in the UA-cam. An electron seemscto be tethered by its attached electromagnetic field lines.
@@christophergame7977 it's a possible account... but are electrons really "tethered" to electric field lines?? What does that even mean? At rock bottom, that "explanation" is simply a heuristic. The Dirac equation for a free field contains no reference to the electric or magnetic potentials. So, for me, that is an insufficient explanation for the physical nature of a spinor and why they necessarily appear in the mathematics of fermions.
@@jmcsquared18 Thank you for your valuable response. You may be right about electrons. But they are just a conceivable physical example of my point about spinors, considered as geometrical objects. There is a geometrical difference between a free rotor and a tethered rotor. Is it fair to say that this is a fair intuitive geometrical picture for spinors?
@@christophergame7977 true, spinors exist outside of physics. But physical intuition could provide some help because, again, we're looking for intuitive guidance which is difficult to come by with spinors. For instance, we don't always say that electrons are just quantum particles with certain charges or spins; we also say that electrons (or any fixed-mass elementary particles) are irreducible unitary representations of the Poincaré group. The mathematical structure of Yang-Mills QFT's gives intuition for quantum particles, and vice versa. One reason we have so much trouble with spinors is that, historically, we found the quantum theory first, rather than e.g., with electrodynamics in which we had a rich history of the vector potential theory (Maxwell) long before it was quantized. Some are researching the foundations classical spin-1/2 field theory in hopes to gain insight into the nature of spinors.
This is the first video I've seen of yours, and it's impressive. I have some background in this, but I was still re-thinking concepts with the help of you're exceptional visualizations and explanations. This is a much better format than just on the math side, and you've done a service to people here.
a well written and animated 1 hour video essay on abstract mathamatical objects and their relaiton to computer graphics and quantum mechanics? i subscribed within about 3 minutes.
It really amazes me that there are people who actually spent time developing whole theories about... how you can rotate something (???!!!). God bless them.
You have that comment on never being able to illustrate quaternions as well as 3Blue1Brown, but wow this video looks great! It shows a lot, but draws your attention to whay you should look at, and is just beautiful and elegant!
Can you imagine thinking about this stuff and working through it back in the day with nothing but pen and paper? No sims or animations, no reference images. Absolutely insane. Thanks for the amazing video 👍
I'm an undergrad student in physics and I just saw the Dirac equation for the first time in my lectures and was so awe struck. Seeing it for the first time, I just thought: "This is so incredible" even tho we just started with the very basics and I don't know the full scope of its implications. I immediately wanted to dive deeper and the alluding usage of the word "spinor" that wasn't explained in my lecture at all brought me here. I am so happy, youtube showed me this as a first result searching for spinors, your video is absolutely incredible. You have sparked a tremendous fascination for this subject in me, which I am very grateful for! Thank you for making this video :)
I've been wanting to understand the spin-statistics theorem, but all the explanations I've come across look very complicated. One of those cases where a simple theorem doesn't have a simple proof, I guess. I might check out that book you recommended.
Hey! :) Yeah, proving spin-stat turns out to be super complicated, it’s a rabbit hole but definitely worth diving into. That spin-stat book was what convinced me that spinors are actually genuinely mysterious. Btw I should have given you a shout-out in this video. Your spinors series is awesome, and very helpful. I’ll be sure to mention your series in the next video!
Topological holes cannot be shrunk down to zero -- non null homotopic (duality). Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Inclusion is dual to exclusion -- the Pauli exclusion principle is dual. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Antipodal points identify for the rotation group S0(3) -- Dual perspectives! Points are dual to lines -- the principle of duality in geometry. "Always two there are" -- Yoda. Duality creates reality! Spinors are mobius loops. The Klein bottle is composed of two mobius loops -- self intersection. The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle. Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates.
Have you explored the interpretations and formalisms for spinors within Geometric Algebra? It's pretty cool, especially the stuff in Projective Geometric Algebra!
Clifford/geometric algebras are some of the most pleasing things I've ever set my eyes on in mathematics. It's like learning complex numbers all over again, except you can do perform operations in high dimensional spaces or space(times) with nonpositive signatures. It's absolutely stunning.
This was my first intro to GA: citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=97f522fa52db9f8ac08bce28f38179e88a67524b In the context of regular 3D rotations, his treatment utterly demystifies the whole topic (Quaternions represent a pair of reflections). And the reinterpretation of the pauli spin matrices is fascinating, but I'm not 100% sure I fully understood it or that it was ever fully ironed out. It raises serious doubts that the usual approach is the best way though.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality). Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Inclusion is dual to exclusion -- the Pauli exclusion principle is dual. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Antipodal points identify for the rotation group S0(3) -- Dual perspectives! Points are dual to lines -- the principle of duality in geometry. "Always two there are" -- Yoda. Duality creates reality! Spinors are mobius loops. The Klein bottle is composed of two mobius loops -- self intersection. The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle. Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates.
Check out the channel xylyxylyx . He has been going through a foundational spacetime algebra paper for the last dozen videos or so. He just got done with how STA represents spinors. It... makes more sense than this... Unless you actually want to do math with it by hand lol. SU(2) is mathematically _much_ simpler, but seeing spinors spelled out in a totally geometric form gives you a much better intuition than the flagpoles and scatter plots.
Just a quick comment on how much I love your channel and really appreciate the amount of effort you must put in to make these high quality videos. I have been trying to understand spinors for awhile now and was so happy when I found your video on it. I can tell you I have watched this video 3 times now and feel like I understand spinors about 50% which is the most I have EVER understood them. Appreciate your time and efforts on all these excellent videos you put together.
That video about the Belt trick and a few others have helped me understand somewhat about what spin represents, but this video is able to initiate people with little math ability to the mathematical beauty and rigor of this mapping of real to complex rotations! Your graphics were a near perfect aid for this purpose! The physical quality of fermions I can intuit now is that they are _tenacious_ little critters. Thank you very much indeed.
If anyone tells me that the internet bill is too high, or UA-cam premium is a waste of money.. I think they don't realize that beeing able to find, watch, rewatch, review, discuss and share this information is so so, SO valuable... Atleast to me. 🤷♂
Great video, and not just because it’s a fun surprise that my animations made a guest appearance! Definitely going to go back and watch your earlier videos next now that I’ve discovered this channel.
I use C++ to write Maya plugins which extend the ways I can generate and deform the geometry. From within Maya I can then keyframe the parameters which drive those custom deformations and render the resulting animation out as a bunch of still frames. I compile the frames into final gif animations or mpeg videos using the ffmpeg command line utility. The animation you showed here is actually pretty easy to recreate. All you need is a way to write a function which can map each given vertex Q in your original mesh to a new position P based on a time parameter T. Here’s the algorithm! ----- First, compute the distance D of the vertex from the center of rotation C. This will determine how much of the rotation should be applied to the vertex. You can define an inner radius R0 within which the full rotation will be applied and an outer radius R1 where no rotation will be applied. V = Q-C D = clamp(mag(V), R0, R1) K = (D-R1) / (R0-R1) Next, compute which direction to rotate the core. Note that the amount the core is rotated is always 180 degrees, and what is actually being animated is not the core itself, but the axis of rotation B which is used to turn the core upside down. θ = T * 2π B = (cos(θ), 0, sin(θ)) Finally, just rotate by some percentage of 180 degrees about your spinning animated axis based on the radial distance to the vertex! Φ = smoothstep(K) * π P = C + V * exp(Φ * iB) ----- In case you haven’t seen it, there’s an animation on my channel which hopefully shows how this works a bit more intuitively. There are a bunch of nested hemispheres with a rubber band stretched across them and a spindle sitting above them. The spindle rotates, the shells turn upside down around it, and then a snapshot is taken of the state of the rubber band. The shells are then rotated back before the spindle moves to its next position. The sequence of snapshots can then be used as a twisting animation of the rubber band.
What a beautifully done video. Your attention to visuals, details (right down to the time stamps at the beginning), your easy to digest script topped with your soothing voice, couldn’t be better 👌🏼 *chefs kiss*
Just came across this video and it completely melted my brain. Beautifully animated, illustrated,explained and narrated, but definitely material for multiple viewings. 😅 Amazing effort on your end. Thank you.
"I want to show you a subtle thing that will open up a crack in reality, which we can use to smuggle this into our imagination" The rawest line that has ever been dropped since at LEAST 1687
As a stranger, I stumbled into a strange land and witnessed wonders to behold. That was a mediation and twas even a glimpse behind the curtain. I am not sure UA-cam’s intention with this one, yet I truly enjoyed.
I am in grade 12...i was just reading resnick halliday krane and came across a spinor problem in rotational kinetics chapter...unable to think anything i came here...needless to say i couldn't understand everything told here😄but i hope one day i will be able to understand it fully.... But thank you sir for ur efforts❤
Spinors are intriguing mathematical entities, defined within structures like vector spaces and Lie groups. These frameworks give them behaviors and characteristics that might not always directly correspond to what we experience in the physical world. Nevertheless, spinors are incredibly useful for making sense of real-life phenomena, despite their abstract nature. It's important to remember, though, that while they provide valuable insights, they're still just mathematical representations and shouldn't be confused with the concrete realities they describe.
@@GetSwifty As English isn't my native language, I've put in some effort to express myself as fluently as possible. Apologies if my writing comes across as a bit robotic! ;-)
I saw a glimpse of this in my dream the other day. No idea what I was a looking at until I found this video! Thank you for all the information and visual representation!!
As a grad student studying topology, I loved this explanation of spinors! The connection between the algebraic properties of division algebras and the topological properties of spheres is one of my favorite stories in math, and it's always fun to see fascinating topological situations come up in interesting ways (the Hopf fibrations lurking in the background). I'm not a physicist, but it is very satisfying to see the physical origin of tools used by some of my favorite mathematicians to construct things I do care about, like Freedman/Donaldson constructing exotic R^4s or, more relevant to this video's contents, Milnor constructing exotic 7-spheres. I hope you continue making beautiful mathematical animations! They are ridiculously impressive.
Is it a coincidence that the alternative that makes computation easiest, SU(2) rather than SO(3), also is kind of chosen by nature, chosen by reality? As if an analogy to The path of least resistance shines through?
I was about to say definitely yes to a video on electromagnetism as a gauge theory, then I saw the three hour video at the top of my recommended! Very glad to have stumbled across this video and this channel, it's already helping a lot of things to click for me. Appreciate the in-depth references, too.
Love the way you explain and show images and equations, making everything way easier to understand. Ive been watching your vids for the past week and got hooked with this channel
Although I have no chance to understand the whole content of the movie, it seams to be extremely well explained and what's obvious for me, extremely well converted into a beautiful and coherent grafic language.
Been patiently waiting for this one. Welcome back Richard. You made up your absence by a literal 70 minute giant, I'm happy.
Glad to hear that! :) Yeah sorry, I would have posted sooner but it took forever to make 😅
I know how hard it is to work with manim, assuming that’s at least partly what you used. And to explain one of the hardest topics ever with it? To the public? With a 70-minute video? ~3 months is honestly an impressive duration. I am sure it’s a great video too, I haven’t finished it yet but that’s my first thing to do tomorrow. Thank you and keep up the good work 😁
@@RichBehielbakers gotta bake
Richard is great, what a beautiful representation of spinors.
@@RichBehiel Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Inclusion is dual to exclusion -- the Pauli exclusion principle is dual.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Antipodal points identify for the rotation group S0(3) -- Dual perspectives!
Points are dual to lines -- the principle of duality in geometry.
"Always two there are" -- Yoda.
Duality creates reality!
Spinors are mobius loops.
The Klein bottle is composed of two mobius loops -- self intersection.
The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
This to me is another stellar example of why youtube may be one of the greatest libraries of human knowledge ever collected in a single place. Congrats on offering up your tome, and for supporting the spread of quality information.
Like in the Time Machine H.G wells🎉🎉🎉🎉🎉🎉🎉🎉🎉
The internet CAN Be used for good.
You tube videos will disappear one day. Information on the Internet is ephemeral.
That too available at all levels of understanding
Hehe if ya love UA-cam so much maybe you should marrrry it then 😊
I recently got a PhD in atomic physics and found this extremely enlightening on concepts that I took for granted all this time. I wish videos like this existed when I was in grad school.
Indians need not apply.
Lets go jackson we love to hear it keep up the good work
Thank you ~ NOCH TSR The BLK interstellar Propulsion Process inventor. Hope that don't go too far over y'all heads
@@june2892 we're not black, so no, we got it probably faster than you did.
@@redflipper992 , I leaned back and thought to myself: "The entity that traveled almost 2 years back in time to deliver me a message about the racism and Human Violations I was going to receive. And why would they travel that far , in salutation for [me] , only to allow kkk , clandestine, Jim crow, etc. to continue stealing, like, for instance, brain waves (why did I allow my Nigerian doctor to hook me up like that?) and other devices utilized for evil?" Then it came to me. These entities are watching too, waiting to kill these wicked thieves something terrible and unimaginable. I felt much, much better.
anyone who's ever tried to plug in a USB cable/stick intuitively knows about having to rotate an object more than 360º for a full rotation
Ahh yes, the superposition of USB 😂
lol, brilliiant, yes! almost always only on the third rotation
I was told USB cables were 4 Dimensional objects... but perhaps they're spinors too?😉
@@markzambelli why_not_both.jpg
I wondered from the first one I ever saw, “how could you start with a DB-9, and get to THIS, and call it better?” Guess they didn’t employ any mechanical engineers…
Wow. This is what youtube should have been. Not youtube shorts that rot my brain chaining me to scroll endlessly for miniscule amounts of dopamine and serotonin. Thank you. Honestly. Thank you.
Thanks Drake, that means a lot! :)
Your are absolutely welcome sir. You have damn well earned it. 👏 I will be checking out your other content feverishly. Haha
You can get it to stop showing you shorts all the time if you tap the three dots on all the ones that show up on your homepage and tap "not interested". It'll eventually start recommending them again, but less and less often the more you tell it "not interested" in every short it puts on your homepage.
Amen
@@RichBehiel true
This is what UA-cam is for. ❤
I don’t normally comment on posts but this deserves a bump in the algorithm. Well done.
Thanks! :)
I know this was for a primarily physics audience, but I have never had SO(3), SU(2), quaternions, and spinors explained to me so clearly in any video ever. As someone from more of a programming background with interest in rotations and vectors from an algorithmic perspective, I've vaguely known about quaternions and matrices and their relation to rotation. But never have I ever had these objects explained in a way that I well and truly understood in a way that I could explain to others. I still am not 100% on the link between quaternions and spinors since you kind of glossed over it here, but I feel like I've definitely taken a major step in being able to get it.
The mathematicians out there should learn that rigor is not explanation! I've seen videos that rigorously explain what spinors are, precisely, and I kind of got it. But I never made the connections on how all the parts really fit together until this video. So thank you! For me, it's all about understanding the motivations and framing the concepts in a way that you "discover" them on your own. That's how you build true understanding. You did an amazing job of that here.
"If you get this concept about the two homotopy classes, if you really feel it, then instinctively you'll suspect that maybe there might be some mathematical object that is sensitive to the homotopy class of rotations... you'll yearn for it"
I can tell you've done an incredible job of setting up the intuition for this subject because that was exactly what I was thinking by this time in the video.
Thank you my friend. I just took the message you sent me. We are going to Evolution
What an amazing setup that was. By the time he named the weird little 4d vector thingy as the spinor, I shouted 'hah' out loud, and my wife gave me the look. Totally worth it!
I wasn’t expecting such a deep/philosophical dive at the end with the spin-statistics theorem. I left inspired after watching the whole video. Appreciate such masterpiece.
Maybe the most interesting youtube video I've ever watched.
As a layman I've always had a casual interest in these topics, this tied together so many things I was curious about but never quite grasped. Thank you for your work!
"A wiggle is homotopic to an octopus" beats "a donut is topologically a coffee cup" six ways to Sunday. Excellent presentation!
My favorite is a human being topologically a donut
@@ExpandDong420 "7 holes" - VSauce
A human being is actually topologically equally to a Honey Comb cereal@@ExpandDong420
I'm sorry you had a poor education.
@@ExpandDong420 No. It is things you can squeeze together to a ball, and things you cannot squeeze together to a ball. Anything that has a hole in it, cannot be squeezed into a perfect ball, the hole will still be there, even if it is smaller, a rift or whatnot.
And so comes the conclusion, there are two shapes, and every item is one or the other. And so the question: Is the universe one or the other, which shape is the universe.
In formality, a priori, knowing nothing about the universe, it's a coinflip. It could just as well be one as the other. The chances the universe is like a teacup (with a handle, that makes a hole in the shape, so you cannot squeeze it into a perfect ball), is 50%, and the chances that it is like the coin that you flip, that can be squeezed into a ball, is also 50%.
The shape with the handle, is what gives birth to the idea of a wormhole. Which we can then say, has a 50% chance to be a possibility, in our universe, a priori, knowing nothing about the universe.
It really goes to show, how farfecthed and thinly veiled in philosophy, the idea of a wormhole is, given that it has become a thing in the imagination, put beside established fact in the minds of man. If the universe has a handle like a teacup, can you then travel along that handle, to show up somewhere completely different than where you started?
This is SO. MUCH. WORK. How did you get this video out the door? My god the animations!! Here's hoping that if there is a day job in your life, it is paying really well. This is waaaaaaaaaay more valuable education than I can get from paid sites.
Thanks! Yeah it was a lot of work 😅 But the ratio of time spent working on it, to the time that people will spend watching it, is a really good deal! It’s a great way to put positive vibes out into the world.
At least one of the animations is stolen from pbs spacetime, the complex 3d one
@Shplinkinshploinkin good point! That animation was by Jason Hise. He published it copyright free, so anyone can use it. I put his name in the thumbnail and later on in the video when that animation comes up. Credit where credit is due! :) I rarely borrow from others, but his animation was too beautiful not to include. I did all the other animations though.
@@RichBehiel Is everything in your video made in Manim? If so that's seriously impressive
@123string4 Matplotlib actually :)
Again, the most pellucid explanation on the topic you cover. Last time complex numbers, and the Dirac equation. This time, spinors. Bravo, Richard. Bravo.
Thanks Curt, that means a lot! :)
Hey what u doing here, curt? 😅
I've learnt a new word, "pellucid" : ) nice
Correct use of the word pellucid gets a bravo from me! I just gave myself a headache distinguishing perspicacity from perspicuity, all the while forgetting that pellucid is euphonically lovelier than either!
The fact that you provide this wonderful masterpiece of a video for free for everyone to see and study says a lot about your character. Thank you so much for your dedication to science and education!
Am i day-dreaming. This is so ridiculously good. You are a grandmaster educator. Thank you.
Thanks, I’m glad you enjoyed it! :)
Guess I found a new channel to binge while I sew. This video was so fascinating.
Here is some intuition on the two rotation types hopefully:
Grab some object ( not needed, you can use an empty hand, but an object makes visualisation way easier )
Without re-gripping the object, If you make a class II rotation, the object will end up in the same orientation ( by definition ), and your hand can also end up in it's starting orientation.
If you make a class I rotation, the object will end up in the same orientation, but your hand will end up twisted, and the only way to fix that is to make a second class I rotation ( or re-grip the object ).
In all cases, you can just make the exact same class I rotation again.
( may not be obvious at first how to do that though )
The two class I rotations together form a class II rotation, which means that your can end up how it started
If you do that a few times with different rotations, there is a 90% chance that you can now intuitively differentiate which rotations are class I and which are class II just by looking at them.
You have also just demonstrated having to turn an object ( your hand ) around twice for it to end up in its original state. This happens because it is connected ( with specific constraints ) to an object which itself cannot rotate.
Ok I’m playin with this but it doesn’t make total sense. It seems like class II rotations I have to move my entire body around the object, like walk in a circle, whereas class I rotations my body stays still and my arm rotates (rather uncomfortably) and then I have to regrip to get my arm and hand back to its original configuration. Is this correct?
@@floydjaggy translations (moving the object) don't matter for this
Yes, doing the belt trick or something similar after watching this video did it for me! Spacetime is sticky for an electron perhaps.
I was just looking to learn more about quaternion and now I have some existential crisis over the fact that their "square root" hold the universe together by preventing some atomic collapse. Great job
I'm sorry you had a poor education.
@@redflipper992No your not
@@devilsolution9781 >your
@@redflipper992you're small.
@@aeriagloris4211 You're brown.
Finally! Someone explains spinors in a way I can understand! People have been trying to tell me stuff like "Spinors are like the square root of Geometry." or "Spinors are these special vectors that satisfy a certain requirement.", or other statements that mean nothing to me. But you. You just simply said "Spinors are vectors with complex components." and it all made perfect sense. I actually think about complex numbers and quaternions all the time, so explaining stuff in terms of complex numbers and quaternions is quite useful for me.
Truly underrated creator. You deserve to be up there with Numberphile, 3Blue1Brown, etc. Loved the video!
Thanks for the kind comment, and I’m glad you loved the video! Hopefully I can be up there with those guys someday. I’ve got some catching up to do, though! Working on the next video now :)
Amazing work! This is such a great service to the physics community to see this discussed so lucidly and with a friendly tone.
Thanks Thomas, that means a lot! :)
Not just the physics community. This is awesome 🤩
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Inclusion is dual to exclusion -- the Pauli exclusion principle is dual.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Antipodal points identify for the rotation group S0(3) -- Dual perspectives!
Points are dual to lines -- the principle of duality in geometry.
"Always two there are" -- Yoda.
Duality creates reality!
Spinors are mobius loops.
The Klein bottle is composed of two mobius loops -- self intersection.
The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
@@hyperduality2838 Thank you 🙏🏻 for the comment. It was stimulating. I liked the parts about spinors, Mobius loops, and Klein bottles from topology. I have thought about these things as definitely related, I didn’t think anyone else had. 😂🤷♀️
Another tip of the hat to your brilliant pedagogy. Without you even trying the first few minutes of your video will surely be the ah ha moment for those struggling to grasp simplicial complexes, vietoris rips and persistent homology. It’s better than anything I’ve seen on UA-cam actually trying to “explain” the subject! Let alone the rest of the video. Just WOW
Thanks for the kind comment, and I’m glad you enjoyed the video! :)
Very nicely put together!
Thanks!! :)
bold of you to assume i clicked on this video without a preexisting lifelong addiction to physics
I'm sorry to hear that
I cried from the happiness that your lecture gave me. Thank you sir.
This is so dope. Like, easily one of the most impressive videos I've seen on youtube. It's elegantly concise, with relatively minimal assumptions about prior knowledge. Plus the whole thing's informed by an simply communicated, profound intuition for the originary principles of topology. Amazing way to frame an introduction to spinors. Not to mention so many amazing labor intensive animations. Awesome aesthetic intuition throughout.
Rare to see so much work go into a video that truly privileges' a generous pedagogy. Been a huge help for me today in creating tangible mental images as I pursue self-education in an intimidating subject. Sure thousands of other folks will feel the same gratitude. To me, nothing's cooler than someone who works so hard to share their hard earned skills like this. Especially when it's given to everyone for free.
Thanks dude!
I was thinking of how does something move from one Planck length to the next because of the discreet nature of it. Your presentation on Spinors here gives me some food for thought! I'm super amateur, but I love thinking about this stuff! I can't wait to finish this video after work and get to the part about the complex numbers. There may be a connection to a thought I was having about higher dimensions and branes. Thank you very much!
I don't know enough to comment on the quality of the content, but this is amazingly presented and VERY interesting. Thank you for the obviously large amount of effort you put into conveying this abstract necessity.
Thanks, I’m glad you enjoyed the video, and I appreciate your kind comment! :)
What a good video. I especially love the audible frustration at around 34:00 about *what, why would you need anything beyond this*, it's a very relatable feeling.
Bravo. Well done.
I was reminded of a mysterious native-Colombian practice that MAY be experimental evidence of applied spinor theory.
What comes to mind is my recent experience meeting a delegation of Kogi (of Santa Marta, Colombia) a surviving native culture with generational teachings based on a deep connection to nature over millennia. My understanding, is that the Kogi rarely travel, but do so as a mission to bring balance to the unbalanced world.
The Kogi taught the guests a practice of communication with nature that immediately resonated with my modest understanding of spinors.
Specifically, when the Kogi communicate with nature (e.g. a tree), with a balanced and open mind, they keep directed ( to and from the tree) intention in mind and spin twice to transact the balanced communication. On learning this practice, I was immediately reminded of spinors.
I am still resolving my feelings regarding this practice and a correspondence between intention (as a real experience described by an exclusive thought pattern) and fermions (as a real particle described by the Pauli-Exclusion principle). My musings have lead me to visualize the local (human) intention and remote (tree) "intention" as two fermions with opposing spins that temporarily entangle (perhaps as a low resistance superconducting Bose-Einstein-esq spin ice?) to transact the communication.
My guess, is the (personal) experiment may be evidenced by noticing a reduction (maybe by half) of original human-intention as a feeling and an increase in an awareness of "tree-intention".
Anyway, this is a work in progress and I felt a need to share...
Thank you for your video and making contact with the mysteries involved.
This is a wonderful comment.
I'm instantly reminded of a video where they were talking about the Native children and how they were more equipped to understand Einsteins theory thru their experience with nature.
Have you seen the Fermionic Mind Hypothesis?
There's definitely something here to your idea. I like this sort of out of the box connection drawing.
You have just given me the extremely rare and thus appreciated sensation of: "Yeah, I know where this is going, I have dealt with this befo... OOOoooooh, that's a neat way of looking at it!"
I’m very glad to hear that! :)
3blue1brown has been real quiet since this dropped. The animations are astonishing and I'm impressed by how efficiently and continuously you were able to explain this difficult idea. Good job
Hes going to become the 3blue1brown of physics
Lmao is it a competition
What is 3blue1brown?
@@nicolasolton hes a youtuber who explains math consepts with great visualizations
3blue 1br is still an absolute master, as is this guy.
this is an incredible video. i'm a computational chemist that regularly does quantum mechanical calculations but i've never had a chance to really peek under the hood of the code i run. this is a great starting point!
Engineer here, run for cover. It seems that the spaghetti animation is showing something interesting that I'm not sure has been made explicit here or elsewhere. The 720 rotation has two phases where the first phase has the spaghetti strand is going over the origin and the second its going under the origin, and that might visually explain the double cover. I've seen that type of animation a good number of times, but haven't noticed that connection until watching this video, and haven't yet heard it explained in that way. No doubt this may be obvious to those familiar and comfortable with the ideas. Thanks for the video, good stuff.
These visual representations that you have created for this video are incredibly good. I'm a computer scientist and not a physicist and yet I feel that you've made it easy for me to follow along with the concepts.
only seven minutes in, but I have to say, personifying the loop is a stroke of genius. No we cannot hurt the loop!
I finished my master's in physics eng. about a year ago and sadly I don't think I will come back to academia after that. But this video is so inspiring it makes me want to be a student again
I love the way this is simplified by the host!! The comedic relief is a blessing as well. And the mystery of the spinors is my favorite part!! I think spinors could easily explain the mental and astral planes in relation to the physical plane. But first I need people to understand these formulas. lol.
This is amazing. Pure youtube gold.
I remember studying group theory two years ago and being confused by the 2 homotopy classes of SO(3) and the fact that SU(2) was it's double cover. Now, thanks to you, I can say that I visually understand that, but the simple confusion evolved into an existential confusion.
By the way, loved the bits of humor included every now and then in the video.
Regarding 44:52. Gimbal Lock is not an issue inherent to SO(3), and using SU(2) does not avoid it. That's a common misunderstanding that gets repeated a lot. Gimbal Lock can happen when you naively attempt to compose a rotation using Euler Angles. It's an issue with the *mapping* from Euler Angles to the space of rotations. It doesn't matter if you use SU(2) or SO(3); if you try to compose a rotation from Euler Angles, you can run into Gimbal Lock.
For real?? Bah, for people like me who have very limited visual imagination these things can be so unintuitive!
The only upside of Euler angles is that they are invertable in the 18th C century.
Interesting! Thanks for the correction!
I’m confused though. Won’t any representation of SO(3) have poles, which would present a problem for a 3-gyroscope system? Whether using Euler angles or axis-angle vector.
Or are you saying that, suppose we had a four-gyroscope system, we could still use SO(3)? I see how we could do that in a roundabout way through SU(2), but I’m struggling to see how to do it directly in SO(3) without the poles being a problem.
@@RichBehiel Tait Bryan angles, AKA "roll pitch yaw" used in aircraft attitude. Since I've written tons of code for rotations in quantum mechanics, remote sensing and mars landings...I've used many reps (and spacecraft GNC ppl use "quaternions" and if you said SU(2) repression they would have no clue what you're talking about).
Anyway, any rep that has 3 different axes shouldn't be degenerate, and requires a computer to invert. I am convinced the only reason Euler (an absolute genius, ofc) used repeated axises, thus allowing degeneracy, is that his iPhone had a lousy chip and he wanted to be able to invert rotations, which for Euler angles is (alpha, beta, gamma) --> (-gamma, -beta, -alpha)..no chip required.
In a related note, doing geodesy, I also learned we have the first, second, and even 3rd eccentricity, flattening, etc..is because early cartographers couldn't do definite elliptic integrals with parchment and an inked up feather.
@@RichBehielI would see the Stack Overflow post titled "Quaternion reaching gimbal lock". The first and second answers are very elucidating.
The way a rotation matrix stores a rotation is not via axis angle or Euler angles but by storing the orthonormal basis that results from the rotation.
This is a really wonderful video. Thank you for making subjects like this so approachable to visual learners.
Thanks, I’m glad you enjoyed the video! :)
I have only an elementary understanding of math and physics but nonetheless I watched this richly complicated and masterfully created video to the end because at a very early moment in it I was struck by the intuition that herein lies something of great value and immense importance to understanding our reality. The quote from the Spin Statistics book at the end confirmed my hunch and I am glad that I persevered to the end though that is not to say that the insights gained along the way were insignificant. Thanks to Rich for putting what to us who are toddlers in his world seems like several lifetimes of unimaginable hard work into this masterpiece of exploration and explanation into the mysterious nature of spinors. His humble and humorous way of presenting his work is very endearing and makes for us babies in the physics world what would otherwise be indigestible content something that we want to roll around in our mouths and suck upon even though we can never hope to bite down on it and really get to the marrow of the matter. So thank you, thank you!
Wow, that’s a very kind comment! Thanks, and I’m glad you enjoyed the video! :)
I don’t usually comment on posts, but this one deserves a boost in the algorithm. Well done!
Thanks! :)
The graphics alone are amazing . Such a lot of work but it does make for a fascinating video .
Excellent presentation! I have a phd in theoretical physics too. It was a long time ago and I am rusty, so I have reverted to be only a little past the denial stage :)
Thank you for mentioning the excellent "Spinors for Beginners" series by the "eigenchris" UA-cam channel in the video description.
I get it! I finally get it! All the talk of a belt, or the Balinese candle dance, or a little cube inside a big cube and connected by rubber bands, I could never visualize. But I can see that a wiggle is the same as an octopus!
You got it! Once you see that wiggle = octopus, everything else clicks into place.
Axions!!! Do Axions!!!! This is so exhilarating- this channel is like 3b1b for advance physics. LOVE IT
21:38 Class I on top is the best loading icon I've ever seen and I need it!
Not just the belt/plate trick.
Respect.
The "belt/plate trick" isn't a trick. It's a fair intuitive/visual account of spinors. A spinor isn't a rotation of a free object. It's a rotation of a tethered object, as illustrated in the UA-cam. An electron seemscto be tethered by its attached electromagnetic field lines.
@@christophergame7977 it's a possible account... but are electrons really "tethered" to electric field lines?? What does that even mean? At rock bottom, that "explanation" is simply a heuristic.
The Dirac equation for a free field contains no reference to the electric or magnetic potentials. So, for me, that is an insufficient explanation for the physical nature of a spinor and why they necessarily appear in the mathematics of fermions.
@@jmcsquared18 Thank you for your valuable response. You may be right about electrons. But they are just a conceivable physical example of my point about spinors, considered as geometrical objects. There is a geometrical difference between a free rotor and a tethered rotor. Is it fair to say that this is a fair intuitive geometrical picture for spinors?
A spinor is a geometrical object, not a physical object. It may or may not give a nice account of some physical object.
@@christophergame7977 true, spinors exist outside of physics. But physical intuition could provide some help because, again, we're looking for intuitive guidance which is difficult to come by with spinors.
For instance, we don't always say that electrons are just quantum particles with certain charges or spins; we also say that electrons (or any fixed-mass elementary particles) are irreducible unitary representations of the Poincaré group. The mathematical structure of Yang-Mills QFT's gives intuition for quantum particles, and vice versa.
One reason we have so much trouble with spinors is that, historically, we found the quantum theory first, rather than e.g., with electrodynamics in which we had a rich history of the vector potential theory (Maxwell) long before it was quantized. Some are researching the foundations classical spin-1/2 field theory in hopes to gain insight into the nature of spinors.
This is the first video I've seen of yours, and it's impressive. I have some background in this, but I was still re-thinking concepts with the help of you're exceptional visualizations and explanations. This is a much better format than just on the math side, and you've done a service to people here.
a well written and animated 1 hour video essay on abstract mathamatical objects and their relaiton to computer graphics and quantum mechanics? i subscribed within about 3 minutes.
I’m glad you enjoyed the video, and thanks for subscribing! :)
It really amazes me that there are people who actually spent time developing whole theories about... how you can rotate something (???!!!). God bless them.
I learned more than from any other UA-cam Video on QM! Best Introduction to Spinors ever.
Thanks, I’m glad to hear that! :)
Understood almost nothing of the physics parts but I'm inspired, my brain got spun for a good reason. Appreciate your Hard Work!!! 😻
You have that comment on never being able to illustrate quaternions as well as 3Blue1Brown, but wow this video looks great! It shows a lot, but draws your attention to whay you should look at, and is just beautiful and elegant!
Thanks, I appreciate that! :) Nobody does it better than Grant though 😅 But I’ll keep trying to improve!
Superb pedagogy. Thanks for making this.
Thanks for watching! :)
Can you imagine thinking about this stuff and working through it back in the day with nothing but pen and paper? No sims or animations, no reference images. Absolutely insane.
Thanks for the amazing video 👍
I'm an undergrad student in physics and I just saw the Dirac equation for the first time in my lectures and was so awe struck. Seeing it for the first time, I just thought: "This is so incredible" even tho we just started with the very basics and I don't know the full scope of its implications. I immediately wanted to dive deeper and the alluding usage of the word "spinor" that wasn't explained in my lecture at all brought me here. I am so happy, youtube showed me this as a first result searching for spinors, your video is absolutely incredible. You have sparked a tremendous fascination for this subject in me, which I am very grateful for! Thank you for making this video :)
I feel honored to have this recommended before 10k views
I've been wanting to understand the spin-statistics theorem, but all the explanations I've come across look very complicated. One of those cases where a simple theorem doesn't have a simple proof, I guess. I might check out that book you recommended.
Love your videos!
Hey I know you, you made that spinors for beginners video 🎉
Hey! :) Yeah, proving spin-stat turns out to be super complicated, it’s a rabbit hole but definitely worth diving into. That spin-stat book was what convinced me that spinors are actually genuinely mysterious.
Btw I should have given you a shout-out in this video. Your spinors series is awesome, and very helpful. I’ll be sure to mention your series in the next video!
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Inclusion is dual to exclusion -- the Pauli exclusion principle is dual.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Antipodal points identify for the rotation group S0(3) -- Dual perspectives!
Points are dual to lines -- the principle of duality in geometry.
"Always two there are" -- Yoda.
Duality creates reality!
Spinors are mobius loops.
The Klein bottle is composed of two mobius loops -- self intersection.
The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Wow , even eigenchris is here !
Have you explored the interpretations and formalisms for spinors within Geometric Algebra? It's pretty cool, especially the stuff in Projective Geometric Algebra!
Clifford/geometric algebras are some of the most pleasing things I've ever set my eyes on in mathematics. It's like learning complex numbers all over again, except you can do perform operations in high dimensional spaces or space(times) with nonpositive signatures. It's absolutely stunning.
This was my first intro to GA: citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=97f522fa52db9f8ac08bce28f38179e88a67524b
In the context of regular 3D rotations, his treatment utterly demystifies the whole topic (Quaternions represent a pair of reflections). And the reinterpretation of the pauli spin matrices is fascinating, but I'm not 100% sure I fully understood it or that it was ever fully ironed out. It raises serious doubts that the usual approach is the best way though.
I love the cohesiveness of GA, it's too bad that interest was pretty low until computer graphics and robotics came around.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Inclusion is dual to exclusion -- the Pauli exclusion principle is dual.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Antipodal points identify for the rotation group S0(3) -- Dual perspectives!
Points are dual to lines -- the principle of duality in geometry.
"Always two there are" -- Yoda.
Duality creates reality!
Spinors are mobius loops.
The Klein bottle is composed of two mobius loops -- self intersection.
The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Check out the channel xylyxylyx . He has been going through a foundational spacetime algebra paper for the last dozen videos or so. He just got done with how STA represents spinors. It... makes more sense than this... Unless you actually want to do math with it by hand lol. SU(2) is mathematically _much_ simpler, but seeing spinors spelled out in a totally geometric form gives you a much better intuition than the flagpoles and scatter plots.
Just a quick comment on how much I love your channel and really appreciate the amount of effort you must put in to make these high quality videos. I have been trying to understand spinors for awhile now and was so happy when I found your video on it. I can tell you I have watched this video 3 times now and feel like I understand spinors about 50% which is the most I have EVER understood them. Appreciate your time and efforts on all these excellent videos you put together.
Thanks Brian, that means a lot! :) Comments like yours make it all worthwhile.
That video about the Belt trick and a few others have helped me understand somewhat about what spin represents, but this video is able to initiate people with little math ability to the mathematical beauty and rigor of this mapping of real to complex rotations! Your graphics were a near perfect aid for this purpose! The physical quality of fermions I can intuit now is that they are _tenacious_ little critters. Thank you very much indeed.
50:21 Damnn, bro pulled this out of nowhere, that's epic lol
If anyone tells me that the internet bill is too high, or UA-cam premium is a waste of money.. I think they don't realize that beeing able to find, watch, rewatch, review, discuss and share this information is so so, SO valuable... Atleast to me. 🤷♂
UA-cam premium gives more vids?
Great video, and not just because it’s a fun surprise that my animations made a guest appearance!
Definitely going to go back and watch your earlier videos next now that I’ve discovered this channel.
Thanks Jason, that means a lot coming from you! :) Your animations are beautiful. By the way, what software do you use?
I use C++ to write Maya plugins which extend the ways I can generate and deform the geometry. From within Maya I can then keyframe the parameters which drive those custom deformations and render the resulting animation out as a bunch of still frames. I compile the frames into final gif animations or mpeg videos using the ffmpeg command line utility.
The animation you showed here is actually pretty easy to recreate. All you need is a way to write a function which can map each given vertex Q in your original mesh to a new position P based on a time parameter T. Here’s the algorithm!
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First, compute the distance D of the vertex from the center of rotation C. This will determine how much of the rotation should be applied to the vertex. You can define an inner radius R0 within which the full rotation will be applied and an outer radius R1 where no rotation will be applied.
V = Q-C
D = clamp(mag(V), R0, R1)
K = (D-R1) / (R0-R1)
Next, compute which direction to rotate the core. Note that the amount the core is rotated is always 180 degrees, and what is actually being animated is not the core itself, but the axis of rotation B which is used to turn the core upside down.
θ = T * 2π
B = (cos(θ), 0, sin(θ))
Finally, just rotate by some percentage of 180 degrees about your spinning animated axis based on the radial distance to the vertex!
Φ = smoothstep(K) * π
P = C + V * exp(Φ * iB)
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In case you haven’t seen it, there’s an animation on my channel which hopefully shows how this works a bit more intuitively. There are a bunch of nested hemispheres with a rubber band stretched across them and a spindle sitting above them. The spindle rotates, the shells turn upside down around it, and then a snapshot is taken of the state of the rubber band. The shells are then rotated back before the spindle moves to its next position. The sequence of snapshots can then be used as a twisting animation of the rubber band.
Based UA-cam comment
What a beautifully done video. Your attention to visuals, details (right down to the time stamps at the beginning), your easy to digest script topped with your soothing voice, couldn’t be better 👌🏼 *chefs kiss*
Thanks, I’m glad you enjoyed the video! :)
Just came across this video and it completely melted my brain. Beautifully animated, illustrated,explained and narrated, but definitely material for multiple viewings. 😅
Amazing effort on your end. Thank you.
Very relaxing after an exhausting day at work.
Wow!! Just blow my mind! In the best of all aspects... Understanding ❤
30 seconds in, liked and subscribed 👏🏽
You put so many fields together in one video. Awesome.
I'm a physics PhD student and this video reminded me of why I started down this path in the first place. Thanks
Thanks, that means a lot! :)
Extremely epic and legendary
"I want to show you a subtle thing that will open up a crack in reality, which we can use to smuggle this into our imagination"
The rawest line that has ever been dropped since at LEAST 1687
BUT BEFORE THAT LETS DO 30 MINUTES ON VOCAB WITH NO REFERENCE TO REALITY AND U NEED TO REMEMBER IT
very satisfying to watch
As a stranger, I stumbled into a strange land and witnessed wonders to behold.
That was a mediation and twas even a glimpse behind the curtain.
I am not sure UA-cam’s intention with this one, yet I truly enjoyed.
I am in grade 12...i was just reading resnick halliday krane and came across a spinor problem in rotational kinetics chapter...unable to think anything i came here...needless to say i couldn't understand everything told here😄but i hope one day i will be able to understand it fully....
But thank you sir for ur efforts❤
Wow a new video about spinors! ok this time ima try understand them
Me too 😂😂😂
Bro, have I been waiting for this, I started almost rediscovering the stuff from scratch, thank you for your effort
Spinors are intriguing mathematical entities, defined within structures like vector spaces and Lie groups. These frameworks give them behaviors and characteristics that might not always directly correspond to what we experience in the physical world. Nevertheless, spinors are incredibly useful for making sense of real-life phenomena, despite their abstract nature. It's important to remember, though, that while they provide valuable insights, they're still just mathematical representations and shouldn't be confused with the concrete realities they describe.
Reads like ai wrote this
@@GetSwifty As English isn't my native language, I've put in some effort to express myself as fluently as possible. Apologies if my writing comes across as a bit robotic! ;-)
I saw a glimpse of this in my dream the other day. No idea what I was a looking at until I found this video! Thank you for all the information and visual representation!!
Your whole channel is a gem really the algorithm is not promoting such good content is disheartening ❤
Awesome video!
Thanks! :)
"life long addiction to physics" is what brought me here!
I...I think I'm in the wrong lecture.
All are welcome here! Embrace the mystery of spinors! :)
Nah dude. Just listen and apply the first principles to everything you can.
The knowledge will be self revealing.
...backs out slowly, apologetically
Oppenheimer Reference
HAHAHAHA I should be studying Economics ye
This was randomly recommended to me and I loved the whole thing. Thanks for putting in the time to make this. Imma check out your other stuff now
As a grad student studying topology, I loved this explanation of spinors! The connection between the algebraic properties of division algebras and the topological properties of spheres is one of my favorite stories in math, and it's always fun to see fascinating topological situations come up in interesting ways (the Hopf fibrations lurking in the background). I'm not a physicist, but it is very satisfying to see the physical origin of tools used by some of my favorite mathematicians to construct things I do care about, like Freedman/Donaldson constructing exotic R^4s or, more relevant to this video's contents, Milnor constructing exotic 7-spheres. I hope you continue making beautiful mathematical animations! They are ridiculously impressive.
When he said "a wiggle is homotopic to an octopus" he unlocked my memory of Henry the Octopus ........ from the Wiggles
Bold of you to assume I'm not already addicted to physics
Is it a coincidence that the alternative that makes computation easiest, SU(2) rather than SO(3), also is kind of chosen by nature, chosen by reality?
As if an analogy to The path of least resistance shines through?
I was about to say definitely yes to a video on electromagnetism as a gauge theory, then I saw the three hour video at the top of my recommended!
Very glad to have stumbled across this video and this channel, it's already helping a lot of things to click for me. Appreciate the in-depth references, too.
Love the way you explain and show images and equations, making everything way easier to understand. Ive been watching your vids for the past week and got hooked with this channel
I’m glad to hear that! :)
POV: you take 4 flintstones gummies before math class
Science is so bad ass
The real foreal…
Although I have no chance to understand the whole content of the movie, it seams to be extremely well explained and what's obvious for me, extremely well converted into a beautiful and coherent grafic language.
This inspires me to step back in awe. Thank you, for pointing me in this direction, you are so right on!