Quaternions!

Thanks a lot for making these videos to help us understand how the higher dimensions can be used to interpret the environment (line, plane, space axis) of our systems (objects to be rotated). However, something you have promised is still missing. I remembered you mentioned that you are going to make another video explaining why a+bi+cj is not of any practical use. Because it is between 2D and 3D? Fractal dimensions? Have you make the video somewhere and I've missed? Or you have explained somewhere already? Please help ^o^ Thanks a lot.

Thanks so much! I hope when the follow-on is fully hooked up it lives up to your high praise :)

3blue1brown

Well after watching all your quaternions prelim vids (a few times) and played with the applet (a few times) I've concluded that were there a Nobel Prize for "Advancing Education in Mathematics" then you would win it hands down Grant (and Ben). Simply breathtaking achievments in this exemplary work work. I played with quaternions a few months back and "sort of" got a feel for them, but now I have a whole new understanding, much deeper. thank you extra muchly. Happy to support top notch work like this forever ... superb and ... will I ever run out of superlatives... :-)

Chris Jennings

Thanks so much Chris, I'm glad you enjoyed it :)

3blue1brown

Glad to have you on board! As to the hypersphere interactive, I'll just say that you should be pleased with where this project is going.

3blue1brown

This is the one that finally gave me no choice but to give back to you for the wealth of knowledge and intuition you've provided me in the last couple years. The animations in this video present and explain projections I've tried to imagine for literal years (after thinking on them for whole summer days from reading about them in Penrose books, and then coming across their mention in lin. alg.), and which I've attempted to draw and program (not my specialty) to understand. It brought tears to my eyes to finally SEE how stereographic projections link dimensions, and to walk away from a single video finally able to imagine de-linking one dimension from itself to let it fall away as a 4th takes its place (just an attempt at describing it) means so much to me. I've never been able to imagine multiple parts of a hypersphere and the closest I've seen to anything like this is the 4D rubiks cube program (I would die for an interactive hypersphere analog). Anyway, I don't mean to ramble, just to emphasize how affected I am by your video. Thank you so much.

Grant, that was amazing. I didn't quite follow moving up from 3d to 4d, but like Randy I've mentally flagged this video for a rewatch a bit later after I've digested a bit more and read up a bit (ill be checking out those resources he listed). I just wanted to say thank you. In every one of your videos I (we) can tell how much care and effort went into it, but even by the high standard you've set for yourself, this video went above and beyond.

Wow, thanks for such a long write-up of notes. There's a lot of good references here I've copied over to my own notes. Perhaps at some point there should be a video on "why quaternions never caught on".

3blue1brown

Really glad you introduced me to quaternions! I wasn't exposed to them (or forgot, yikes!) in my undergrad physics degree, and I think they give a helpful perspective on both complex numbers and vector calculus. I like the historical perspective and think it would be complemented well in the introduction by a quick placement in the field of mathematics, i.e. where one would encounter them (first course on complex analysis?) for those of us non-math majors. I think it's worth mentioning octonions at the beginning too, and Clifford algebras, which I also had not heard of and know nothing about except the first sentence of the wiki entry "In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems." That seems too weedy but connecting quaternions down to complex numbers and up to more advanced/abstarct math I find very motivating. The brief intro to group theory in your longer Euler video was one of the things that got me hooked. Re quaternions, the fact that Maxwell's equations were originally written using them is fasicanting, and their subsequent rejection by the physics/education establishment (I think at least for the undergrad level) is super interesting. It seems like a mistake to me. I've gone down the rabbit hole and just found another ref related to this: Quaternions in Univeristy-Level Physics by M.E. Horn (2002) "As an expansion of complex numbers, the quaternions show close relations to numerous physically fundamental concepts (e.g. Pauli Matrices). In spite of that, the didactic potential provided by quaternion interrelationships in formulating physical laws are hardly regarded in the current physics curriculum. In particular, many approaches emerge that are useful in conveying the unity of seemingly distinct theories in a didactically convincing manner." Related to this, in the chapter on Algebra in the Feynman Lectures on Physics, "it turns out with this one invention, just the square root of -1, every algebraic equation can be solved! ... the greatest miracle of all is that we do not [have to invent again and again and again] ... we are finished inventing new things." What about j and k? Some qualifying phrases would have been appropriate there, and maybe a mention of quaternions. Even the great Feynman seems to be guilty of misleading students in a effort not to confuse them. On quaternions, I also found this paper helpful: From Counting to Quaternions by J.M. Hersh (2011) " ... octonions are the last step – there are no other division algebras with real coefficients. This may seem surprising, but it makes sense. To achieve each extension to a higher dimension, we had to give up some nice property of numbers. We started in one dimension, with the real numbers. They can do the four operations of arithmetic, and they also are linearly ordered. Next, in two dimensions, we bring in complex numbers. They still satisfy all the laws of arithmetic, but they no longer are ordered. Going up to 3-space with the quaternions, we had to give up the commutative law. To go up to the “octonions,” we must even give up the associative law. That’s the end of the line. After that, all larger “alge- bras” must include “divisors of zero.” It’s no longer possible to divide by everything except zero." And from blurb on book Visualizing Quaternions: "gentle introduction to their four-dimensional nature and to Clifford Algebras, the all-encompassing framework for vectors and quaternions." Maybe some of this is more appropriate in the show notes, where I appreciate seeing "how to learn more stuff". Overall regarding the video, quaternions were so new to me that while I like the animations of the projections, and think they are an important contribution (I'm trying to imagine this being attempted by a textbook, or prof at a blackboard - ugh), I must admit that I didn't fully follow them when you got to the meat of the 4D>3D projections. Which is OK, I've flagged it to come back to them later, in conjunction with understanding the calculations. So, I'm really looking forward to the next video!

I like the format he used at 10:26 of this video. I admit, he probably should have used it again at 26:30 to clarify where that point was located. I like your animation. I never really internalized the infinite points a 2d point could map to.

I use quaternions when dealing with 3D graphs but I have never had appropriate picture describing what is really going on. Unfortunately, I am not that bright so I am going to have to re-watch the video multiple times before I have any real intuition on the subject. Looking forward to the next video, keep up the good work.

Fantastic, I wish I had these videos when I was going to school. I had a natural intuition for algebra, trigonometry, and calculus but was missing that for a lot of other subjects in math including quaternions.

There is actually one more thing that has been tormenting me: representing a point moving in 3D space is very confusing. Without any reference it is impossible to get a feeling about the plane, or even quadrant in which it is located. It simply floats is a void. I see that you continually rotate your view in order to counter this, but it is still, quite elusive! Perhaps one way would be to "follow" the point with some reference projections? I cooked up a GIF because this is easier to show that a point in 3D needs some "anchoring". The exact same point can correspond to any point lying on the ray connecting it to our eye, and that is the main issue. Here is the visual: <a href="https://i.imgur.com/Ow0yW8W.gif" rel="nofollow noopener" target="_blank">https://i.imgur.com/Ow0yW8W.gif</a>

So Felix is basically experiencing an acid trip when the sphere is rotated

Peter Kransz

Fantastic work as always! This video has been do far the most difficult one for me to follow, out of the whole 3b1b library. At some point you mention that a 3D coordinate system does not have good multiplication rule but that is a topic for another video. Somehow though, I feel that this would be a key insight as to why we even *need* quaternions and what made Hamilton's quest futile in 3d. I might say that in general, there was more "let's just take this as a given and move on" than I am used to on your videos, which normally seem to cover all logical steps without leaving gaps. Not sure what to make of this though, since the video is already half an hour long... Just my two cents + immense gratitude for enabling so many of us to grasp previously scary concepts!

One more thought in the hope of making it easier to empathize with Linus. At first the linelander's world is depicted horizontally (3:00), but then we give him a stereographic projection onto ⅈ (8:03) which is vertical. This might seem like a nit-pick, but if you rotate his projection (or ⅈ) to be along the horizontal plane it would simulate a horizon (1 unit away) where Linus could imagine himself spinning around to look in each direction with a super-fisheye lens. It wouldn't be so different to being eye-level with the ground in an early version of microsoft flight sim (where the ground was almost entirely flat: <a href="https://imgur.com/pPKt16G" rel="nofollow noopener" target="_blank">https://imgur.com/pPKt16G</a> ). That's much more intuitive than the video's outstretched noodle-thin spinning space donut in the imaginary plane. It would also mirror Felix's world, where the ⅈ-axis is also horizontal. So it would be more intuitive and make the video more self-consistent to always orient Linus's world horizontally.

Yup, accidentally forget to give that guy the right update function, I've fixed it now though.

3blue1brown

Thanks for letting me know, I'll fix it :)

3blue1brown

Also (though this is an entirely pedantic and trivial observation) you are not using a standard color-scheme on the rubik's cubes toward the very end of the video. A standard cube has the following pairs on opposing faces: red/orange, green/blue, yellow/white. You have them as red/white, green/orange, and yellow/blue. This is not wrong for purposes of the video, but it will nerd-snipe the cubers in the audience and distract them away from what you're saying. (oh, and the opposing faces are arranged such that if you're looking directly at the blue/orange/yellow corners, the clockwise acronym they form spells "BOY".)

jason black

At around 26:45, where you're showing multiplication by an arbitrary quaternion, shouldn't the perpendicular circle (the one in shades of gray) also rotate right-hand-ruleishly around the direction of that arbitrary quaternion?

jason black

This video seems a bit fast pace- I would like to see it divided somehow into 2 parts if possible. $0.02

I also want to raise another topic: tying this back to the essence of linear algebra videos. I think having it stand on its own is important to new viewers but, having invested in watching the essence videos several times, I selfishly would have liked to see the language of linear algebra used throughout. Perhaps it'd be possible to anchor it back more often, with direct links to videos could be possible without alienating those who haven't spent as much time with linear algebra?

Very excited to see this topic getting the 3b1b treatment, as I'm currently grappling with grokking quaternions. The core of the video, imagining trying to perceive something in an unobserved geometry with the help of projections is great, and makes me excited to tie this understanding back to projective geometry. What I had a harder time with as the video unfolded into higher dimensions was following along. It will require repeated viewings, and that's ok, but thought I'd share my initial experience. Other's comments about slowing down or repeating could help.

I love that you're tackling quaternions. In years past when learning 3D graphics programming, I've dug around for a video that does exactly this so I'm happy to see it. Here's my notes. - part 1: This projection looks like Linus is standing at the center of an enormous spinning hula hoop. I'm pretty sure the projection works out the same. Maybe thinking of Linus this way would help in an intuitive explanation of how the projection works? - What's Linus getting out of this? I get that it's showing a 1D projection and it's setting up later parts, but I didn't see how this projection was helping Linus any more than if you just unwound a string from a circle into a line. Maybe I wasn't looking at the numbers on the line closely enough. Can Linus do other calculations or operations on this 1D projected circle or does it just give a visualization? It feels like you're just spinning around a 1D circle and Linus says "oooh pretty!" much like my reaction to a spinning hypercube. How is this a useful tool for Linus beyond "just" visualization? Can he measure the sides of a spinning square (or other object) maybe? Or at least visualize the rotation of 2D objects? - part 1 still: This chapter also seemed to go long, just spinning the circle around and around. Maybe you could compare it to some other way of projecting or unwinding the circle and show what intuitive concept Linus can grok from the projection? Or perhaps you could start with Felix's section, which is more intuitively motivated (projecting a sphere onto a 2D surface is something humans commonly do), and then go show Linus's part second as a simplification of it, like "and here's how this would work in 1D". I realize this wouldn't be as neatly organized, but it might make more sense to naive viewer because 3D→2D is more "obvious" than 2D→1D. (Seeing as you start by showing 4D anyway, you could even reverse the order of the entire video?) - part 2: stereoscopic projection is used to store panoramic photos or can project world maps. Why not (at least briefly) show something other than grid lines to help the intuition? Maybe not mathsy enough? Just a thought - part 2: (10:43) I'd appreciate if you included the reason why "such 3d number have no good multiplication rule" even if it were a single sentence which still needed another video to explain more. It comes across as if you just haven't thought of how to summarize it yet. - part 2: Felix is depicted sitting outside the sphere. However the projection is basically the same as if a very (impossibly) wide-angled camera was sitting at the center of the sphere. Maybe it would help to think about it this way, putting Felix inside the sphere, rather than only thinking in terms of mathematical projections. - part 2: "turns the ij unit circle into the 1i unit circle" (?!) These rotations are very simple concepts, and visually easy to follow, but the precise descriptions you give have a very high cognitive load. It would be great if you briefly highlighted _only_ the ij unit circle as you mention it, then briefly highlight the 1i unit circle, and then show the rotation. Maybe you should start with the rotation about the origin first as it's the simplest. Perhaps name each of the rotations ("pitch, yaw, and roll"?) before you step through them in detail? Or even mention the names of axes and that you're going to show a rotation of each. (You could even put a 3d pi-person-mobile inside the sphere and show how it projects to give a better idea of how each rotation is different?) This could be a good time to mention Euler angles and gimble lock too, although I guess you're aiming for a fresher approach that doesn't rehash the old ground. - It seems you're talking about normalizing quaternions at the start of part 3 (I think?). Shouldn't this be left to after they're more fully explained (visually)? - at 4:30 you could reference/link your linear algebra video which explains this is more detail. Oh, you mention another video. Maybe it's better I don't know - going back to "What's Linus getting out of this?" I feel like I have the same question for the other parts too. I feel like you need to move beyond n-dimensional reticulated spheres and show the projection of other objects (even if it's just more cubes, but preferably things that make more intuitive sense to rotate like a pi-person-mobile), so it's more like "here's this rotation applied to [all the points of] this object". Because trying to imagine a rotated sphere is difficult because it always still looks like a sphere. Maybe you're saving trying to keep the maths simple and keep everything with a magnitude of one, but it sometimes feels too smooth or simple. - Without knowing about the Alice in Wonderland part, I felt like you had the right amount of historical set up. But after reading it, I definitely think it should be included (and perhaps even referenced again later) Hope these comments are too late or too long and hope you can find anything in them to help.

Great points! I'm putting them on my todo's. As to the last one you made, I wholly agree, and I think you will very much like something being cooked up as part of this project that's coming in a few weeks...

3blue1brown

I feel you could do more to motivate the video. By the end of the intro (2:30) I feel like you should have mentioned gimble lock or some other problem to be solved, rather than just relying on the audience to be interested in quaternions for their own sake. e.g. "Quaternons will solve this problem, and we'll see along the way how elegant they are"

First a few nitpicks that I didn't see already mentioned: - At 19:36 the sphere is missing the -j label for no obvious reason. - Also, when you show the projected unit sphere through 1 and -1 etc at 21:20, several labels get removed - I'd find it helpful to still show them that are on the discussed sphere, i.e. -i, i, -j, and 1. - This was mentioned before but I think it's important: the last example at 26:45 really needs to visually rotate the white&amp;grey circle too according to the right-hand rule. It's essential to understanding to not get confused by inconsistencies. Secondly, I'm thinking a bit of the time you give the viewer to look at the animated examples. There's a lot going on on the screen at the same time - the simultaneous rotations and the coordinates changing in the corner, and on top of that trying to get a grasp on the still unfamiliar stereographic projection and the interactions between i, j and k. I needed multiple viewings to fully understand what's going on. Perhaps allow the animations to run 10-20 seconds longer or so? But I can appreciate this is very tricky to time well. (If I were to dream a bit it'd be fantastic to be able to "pause" the presentation and slide things back and forth myself until I feel I got it under control, then let you continue. That's of course an entirely different beast than a Youtube video, but it'd be an awesome educational tool.)

Martin S

Hmm, interesting, I'll give it a try. I believe my thought was that there is more information in the 3d view, which might be hard to see in the smaller view, but I'll give it a try.

3blue1brown

At 13:00 you do picture-in-picture with the planar view being in the small picture and the spherical view being in the larger. Can I suggest swapping that? It's the planar view that we 3d-landers need to grok and I think having it as the dominant view might be helpful. (Or perhaps my mind is just weird.)

Bob Dowling

I agree that the history is interesting - but I'm not sure if it should be separated out into its own video. This is one of the harder (for me) videos that you've done and I hesitate to make it longer. Looking forward to the sequel.

I also found the circles bit to be very useful in understanding what was happening. It was interesting to see the infinite lines close up and gradually become the circles, and vice versa. I do agree with whoever previously said the point about any circle going through -1 becomes a line was for me.

I think that extra history but is quite interesting, and the extra time it would take to say that isn't that long compared to the length of the video already. I think you should go for it.

Great video. One connection I found helped me too better understand quaternions when I was using them in CG was via Clifford algebra and their interpretation of ijk as the three possible oriented planes you can define in three dimensions. That works for any dimensions and help explain how you would rotate in 2d, 4d why thinking of an axis of rotation is only working in 3d as it is the dual of the linear combination of oriented planes etc.

Hi Grant, thank you so much for this sneak peek! I think an extra minute or two of history wouldn't hurt at all, given how long the video already is. The scene with Felix viewing just the three reference circles really made the 3d projection click for me, and by extension helped me understand the 4d projection better.

From 26:45 shouldn't the circle appear to rotate, in accordance with the RH rule? The maths must be tricky, but I'm sure you're up to it :)

Chris Jennings

At around 13:23, another yellow line appears, that seems to connect the projected location of the pink dot to...something. But, it is a bit confusing, because up until that point in the animation, the only yellow parts are the equator on the sphere, and its projected circle. Also, I just noticed the same yellow line doesn't appear in the inset animation on the top right.

Hey Gabe, for sure the next video will get into the applicability to graphics et. al. so I’m glad to hear you’re eager for it. And don’t worry for real there will be a followup. I promise this isn’t like those other times - it’s already well underway and I’m really excited for it! As to the unit sphere changes - I love the suggestion, but previously found it trickier than it should be...perhaps more play here is in order. Thanks for the feedback! I always love this part of the video process - feels like showing it to my friends if my friends weren’t all super tired of me talking math at them while we’re driving.

3blue1brown

Yeah, the mad hatter fact was really interesting to me when I first came across it.

3blue1brown

Thanks so much Andy, good comments to know, I’ll think about how to clarify the lines-are-circles point, either visually or narratively.

3blue1brown

Not sure if your joking, that’s just my hand filmed on a green screen :)

3blue1brown

Hooray, Grant! These visuals give me a powerful new tool to understand what's going on here. An earlier poster asked why ijk = -1, I visualized the rotations, and I saw it immediately. That's new! Regarding your questions: I would hesitate to add more historical background; I found the scene in which Felix views the three reference circles very helpful and not confusing, except that I had to strain and rewatch to understand the point "in general, any line that Felix sees comes from some circle that passes through -1." (14:13). To bring more people along there, you might need to unpack a bit more. Very excited for this work!

While I am not very fond of introducing a lot of historical facts, I am a fan of Alice, so a reference to the Mad Hatter would be just fine for me!

Daniel Armesto

Very intense and hard to follow ... but that´´´'s what makes it interesting and fun!

Daniel Armesto

I'd definitely like some more quaternion history at the beginning. I just now read up on the Mad Hatter's connection with quaternions, and it's pretty fascinating. I'd quite like to learn more, and I think it'd be neat to mention the method of visualizing quaternions with the real numbers representing time. Even without all of this though, it's still a great video through and through. Thanks for making my day yet again, Grant.

Several years ago I had to use 3 coordinate systems (polar, 3d and touch screen) and feed result to the Open GL Camera function. unfortunately the 3D engine back then suffers great deal of bug when the calculation is near the boundary and causing so many lines broken apart.

I guess computation will be more accurate using this Quaternions for 3D engine Open GL Camera since projection becomes rotation operation in Quarternion domain yet the result is projection in 3D domain. (I probably need more wine lol).

Very nice! Two comments I think going into the advantages of quaternions for computer visualization would be really interesting. You mention it is more efficient to compute, but showing this and talking about other advantages would be cool. Second, I think the change in orientation of the unit sphere after a half rotation might be more clear with a visual cue. Maybe color the two sides of the surface differently, so it is obvious when it is inside out?

Gabe

Superb work Grant, just beautiful. I played with quaternions in 3D rotations a few months back. They were tricky but rewarding - so I'm really hanging for the next instalments. Re adding more history: one of your goals is to promote a love of maths, and I strongly feel that includes history of maths. Doesn't matter a hoot if the video gets a bit longer, we all loose our sense of time anyway, as we get hypnotised by your visual maths choreography :) Noticed a minor grammer issue at 10:43 suggest it should be the plural "Such 3D numbers ...."

Chris Jennings

I think the history interjection should be in the video. It's a long video, so making it a little longer won't break it. BTW, I love this. I am super excited about this. I love imaginary numbers and knowing more if their history. I'm am very much an autodidact; are there any books you would suggest on this subject?

Hi Grant, I'd just like to know how much time you have spent on the 3d hand for the right hand rule. This is not some geometrical object you can easily generate with a short piece of code, right?

Totally agree. I feel it would help build intuition much more, even just spending a few minutes more in Felix's shoes alongside the 3D motion.

Ooh, good suggestion.

3blue1brown

Oh interesting, I didn't think about using that as an example. Think of it i(jk). Point your thumb to j, and notice that it rotated k to i, so jk=i. Now you have i(i), which is -1.

3blue1brown

Does quaternion conjugation relate to linear transformations in an eigenbasis?

Noted on the music. As to the axes, I originally flipped all of space, but then realized I wanted to keep the displayed plane in the xy direction, and fading out the old plane and in a new plane seemed a bit odd. I can see an argument either way, so I'll think more on it.

3blue1brown

It seems that the video tries to explain stereographic projection more that it does explain quaternions, honestly. In that vein, to help explain stereographic projection, I think that some representation of what the real component looks when doing quaternion rotations like would also help, possibly a 1 D number line accompanying the 3 D sphere?

mazterlith

Since I was first introduced to quaternions through computer science, I always thought of the values of the quaternion as linear combinations of a 3 dimensional rotation. As in, the real part was unrotated, i was rotation in the x axis, j in the y axis, and k in the z axis. For a 2 dimensional creature you would therefore only need 2 values for a linear combination to represent 2 dimensional rotation: real is unrotated, imaginary is rotation in the x axis.

mazterlith

Thanks, good feedback to know. It's interesting to think of high information density as having such a benefit.

3blue1brown

Good catch, and good suggestion!

3blue1brown

There's just so much that's interesting here. I could easily do five videos just on quaternions and places they show up.

3blue1brown

Oh yeah, doing this in VR would be extremely satisfying. As to the non-commutativity, perhaps that does deserve a bit more of an emphasis.

3blue1brown

I certainly agree! Which is where the second video comes into play.

3blue1brown

I love the very physical motion of pulling and rotating all of space. I research VR [at Stanford, hey hey!] and I feel like this is one of those things to really get your hands "dirty" with. I wish there was more opportunities for the properties of quaternions to "fall out" of this visualization technique. I didn't really follow the non-commutativity, though a simple "pause and ponder" may be sufficient there. Though to be fair, that may the focus of the next video.

Wow. What a mind*@#$. I'm gonna have to watch it a few more times. For now, I'd just ask you to pay attention to the volumes. As you were switching from Felix the Flatlander to whatever the 3rd part was, it got really loud before your voiceover began. I'm very glad to be learning about this, but some concrete examples might be good. I mean, I know graphics folks use them in the abstract, but seeing what exactly they use it for might help me grok this stuff better.

Burt Humburg

Testing testing.. wrote a long comment and it vanished :( how about now?

issa

Testing testing.. wrote a long comment and it vanished. :( Does this persist?

issa

Hey Grant! Great video, but was a little lost towards the end. Maybe a recap describing how we went from linus to felix to you may help better connect the topics. Good work!

And also, these are your best 3D graphics to date—I'm sure that took a lot of work. It's excellent.

issa

I think the history is neat, and not terribly long, though I bet you could condense the bridge story out of it if you wanted—the essence seems to be the notion that 3 no worky and 4 was a big insight. I found that it was tricky the first time through to remember that there are _two_ operations happening at all times: one is the projection, which is a visualization tool, and the other is the rotation, which is what we are actually interested in studying. After a while of tracking mapped points/lines/etc the two sort of conflated and I had to pause and detangle the two again (what does a point in projected space "mean"? Oh—it's just a point, but it was mapped from some point on the higher-dimensional object, and it's the _movement_ of that mapping that is the _actual_ rotational transformation, which is what we are interested in.) Maybe this is just me—I often end up watching your videos 3 or 4 times before I feel comfortable with them. I also almost feel like seeing, especially near the end of chapter 2, more of _just_ the projected version performing draggy rotations _without_ the original (especially since in the following chapter we indeed don't _get_ to see the fourth-dimensional unprojected original) would be useful, because it might help reinforce that the reason we are interested in this projection is to build some visual intuition for what higher-dimensional rotation looks like, and that this intuition relates to the higher dimension but has some kind of physical logic of its own.

issa

Could you provide more insighy about ijk = -1?

While the additional history is interesting, I prefer the shorter intro.

Alia

Great, others seem to agree, I'll add that in!

3blue1brown

This is actually beautiful but for me it was very hard to visualize what quaternions are. The things I take away are how nice you can think of multiplication of complex numbers and how you can attempt to visualize something via stereographic projection. I think I lack the visual intuition to actually have i picture in my mind, so maybe others will feel very different about this video but in my case I'll have to stay in 2 or 3D for now :(

Supreme

Starting at 12:52 you had an upper right hand rectangular view of the 2D view without the sphere as a distraction so you could choose to look at the simplified 2D view without worrying about the 3D sphere. Later at 13:34 you start showing other great circles on the sphere and how they transform into circles and switch places through rotations and I think that segment would also greatly benefit from a pure 2D view in a box in the corner so the viewer can look at what's happening in 2D without all the overlayed sphere details distracting them.

This is great and I'll be watching it several more times coz it's a lot to take in, at quite a pace. During chapter 2, I was wanting to see more of the view Felix sees in 2D alongside the 3D animations.

Steve Chantry-Taylor

At about 14:38 you say something like "an extension of what Linus the Linelander sees". At first I didn't realize, that you could see the same morphing on the green i-Axis as we saw in the 1D chapter earlier. Maybe you'd like to add "as you here see on the i-Axis, when rotating the green unit circle." I do like some historical context, as to where mathematics, 3D algebra and the like wer, when Quaternions were invented, but I'm unsure whether they should be included in the video itself. Maybe you can add this in the description?

Hendrik P

Hmm... good to know, I'll think about how to rephrase that.

3blue1brown

Thanks!

3blue1brown

My view is that a video should be something I intend someone to watch in one sitting. I felt like if I split this up, each part would feel incomplete an inconclusive. Linus the Linelander on his own is not too interesting, and by the time one gets to quaternions I want to be sure the 1d and 2d analogs are fresh in ones memory.

3blue1brown

Very much enjoyed it, though it took an awful lot to process. I’m curious whether you tried to split this into smaller video chunks and couldn’t, or if you feel like 25 minutes is a sweet spot for explanations of this depth.

Jacob Mirra

When working with robotics, I used to work a lot with quaternions. My general strategy, when using C++'s Eigen library, was to create a Quaternion from an AngleAxis, and then use that to make all the transformations.

Woo! Quaternions! I starting to grasp a visual model in my head now which is great but I'll need to rewatch partsa lot to develop it haha. I don't think adding history is a bad idea, even if it is long. The history can help with the understanding of how things are today and the length is just a reflection of the difficultly of understanding quaternions. If you are concerned then a separate appendix video may be the way to go...

At 10:40, small typo where it says "Such 3d number have no good multiplication rule" should be "numbers" I think

Mind blown. Will replay many times. Great subject. Great explanations.

At :13:50, "Most points on the *actual* i-axis, are completely invisible to Felix". I had to replay this section several times before I understood you, because I didn't know what you meant by the "actual i-axis". (I now understand you meant, the i-axis in the 3D space. It's confusing though because Felix clearly does see *an* i-axis, but just not the one that is in the 3D space.)

Man, I wish I had your video when I had to apply them in software for some rotation tracking thing based on a quaternion and Madgwick's algorithm (look at it, it's actually rather ingenious). I have a few suggestions: Number one is that the background music is a bit too loud when you change chapters. Also, when you flip axes at apporx. 10:15, why not rotate the whole 3d space instead of swapping the labels? That looks more natural and less confusing.

Great video! I am wondering if Linus's segment would not be a little bit better if 2d space and 1d projection had the same amount of screen. It looks a bit cluttered on the right and empty on the left.

I work on a product that has an IMU, and I've never really looked into what exactly the quaternion is. Thank you for starting me down the road to understanding it.

At about 10:51, there's a momentary flash of some other frame on the screen, just as the equation is becoming visible.

I think it's fantastic. I think the history you presented at the beginning is just enough to give you context, without distracting from the subject at hand. And that context is very important to have - I understood the _reason_ for quaternions much better when I came across that story myself. As for confusing visuals for Felix, I think think they're necessary and very helpful as a bridge to your 4D/3D visuals. It helps to be able to grapple with those confusing distortions in a space I can visualize before I grapple with it in 4D. I wouldn't worry about them being too chaotic - I think having that density of information is really helpful. And I can always pause and rewind. And like someone else said, the finger-dragging metaphor is incredibly good. I don't think it would be as effective without that little visual of a hand. I'm not sure why - maybe it helps me picture myself dragging and stretching space? Whatever the reason, keep on doing that. It works real nice.

Ben Visness

In 26:47 the circle isn't rotating! It's really informative. It really gave me a concrete way to think about it. I'd like to have more discussion about commutativity-- why is it good (useful) to have it or not have it. Also why &lt;=2d has commutativity and &gt;2d does not. I mean, sure, algebraic discussions isn't the focus of the channel, but it's still something that's helpful to help with the visual representations.

Liz Av

I think this is a great start! It would also be fun to have a series on the Caley-Dickson construction to see how you can build up the division algebras (and beyond) starting with the real numbers. I'd love to see you explain how two copies of the quaternions can give the octonions. Another very cool application of the quaternions (besides having the same algebra as the -i times the pauli matrices, ie SU(2)) is that they are also integrally related to Lorentz geometry and special relativity. Both quantum mechanics and special relativity are deeply connected to the 4D division algebra :)

Ah yes, not sure what’s going on there, but it’ll be fixed!

3blue1brown

You probably know this, but the blinky pink pole is pretty distracting in the middle part of the video where the sphere is turning around.

Jacob Rus

Absolutely fantastic video! Mistake: At around 27 Minutes, where you show a rotation around an arbitrary axis, you forgot to implement the perpendicular rotation. In addition, while I liked the three circles you did with Felix, I think it might be beneficial or less confusing to show the two-way view (the one with the solid plane you showed before) instead, or at least as well, in the upper corner. I find it a more intuitive visualization, personally, as it shows me the entire curvature of the plane.

Sascha Baer

As for that extra segment, I'm thinking it could do good to add it in, just to add a bit more perspective to it. If the video gets too long, it might be worthwhile to turn it into a small series of shorter videos instead, encouraging viewers to pause and ponder a little more between videos.

Alien Valkyrie

I have to say, this is the first actual visual intuition for quaternion multiplication I've come across. Gotta sleep on this and see how it extends to scaling with non-unit quaternions.

Alien Valkyrie

The reason I was hesitant about doing that is that the character Linus has, definitionally, very bad spatial intuition. So of all the places to conspicuously diversify gender, I feel like Linus and Felix's positions would be counterproductive.

3blue1brown

I really like how you use Flatland analogies to explain the extra dimensions! Few notes here: Maxwell Equations were originally written in quaternion notation. He was using quaternions to describe Oersted's discovery about a special transient force that only makes things (a compass) 'rotate' without displacing it.

Looking good. I'd have spent a bit more time on the dot and cross product version that lets you just use the same (much simpler) math of complex numbers and 3-vectors, but I expect more people have managed to learn things in terms of matricies and linear algebra. I always found matricies horribly redundant and painful to think about but you've certainly made that notably easier for me to think about. It's a bit off topic but, aside from being less redundant, complex numbers, quaternions and their hyperbolic cousins can be used to beautifully derive and use special relativity.

Wow! This is your best yet. I'm going to have to watch it several times.

Doug Fort

Love it overall. The finger-dragging metaphor and corresponding animation for the action of one complex number on another is wonderful. You, might consider changing the name "Linus" to a conspicuously female name and using "she." I know it might seem like a small thing, but we would benefit from attracting more women to STEM (and especially to physical sciences and math), and small signals like that may, in aggregate, go a long way in changing cultures and perceptions in this regard.

awesome coincidence, i am currently learning about them. great timing :D

:) :)