Visualizing quaternions (4d numbers) with stereographic projection

How to think about this 4d number system in our 3d space.
Part 2: • Quaternions and 3d rot...
Interactive version of these visuals: eater.net/quaternions
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/quaternion-thanks
Quanta article on quaternions:
www.quantamagazine.org/the-st...
The math of Alice in Wonderland:
www.newscientist.com/article/...
Timestamps:
0:00 - Intro
4:14 - Linus the linelander
11:03 - Felix the flatlander
17:25 - Mapping 4d to 3d
23:18 - The geometry of quaternion multiplication
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
Music by Vincent Rubinetti:
vincerubinetti.bandcamp.com/a...
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Reddit: / 3blue1brown
Instagram: / 3blue1brown_animations
Patreon: / 3blue1brown
Facebook: / 3blue1brown

[@ignaciodemiguel3683]

My teacher 5 years ago: i is the only complex unit My teacher now: jk

[@thesenamesaretaken]

Underrated comment

[@o5v3]

You win this comment section.

[@Syuvinya]

*Cries in octonions*

[@Syuvinya]

btw you can't spell "octonions" without "onion"

[@urasgungor9244]

brilliant!

[@danielkaufman556]

Now it's 3Imaginary1Real...

[@3blue1brown]

Time to insert some subtle reference ijk being blue and reals being brown in the follow-on...

[@zazkegirotron]

Spoiler Alert!

[@ethandandu]

Next time: 9 Imaginary 0 Real

[@garrytalaroc]

Niooocce!

[@WhattheHectogon]

*dies*

[@gexandthecity7623]

The pure dread of knowing you barely understand the 3d part of the video and know he's going to start speaking in 4 dimensions has more tension than the shining. I would watch a horror movie made by him

[@MarloTheBlueberry]

I barely understood the 1 dimension

[@Fangamer1254]

​@@MarloTheBlueberrySame

[@sonderevokingbuns]

"What you're looking at right now is something called quaternion multiplication; or rather you're looking at a certain representation of a specific motion happening on a 4 dimensional sphere being represented in our 3 dimensional space. One which you'll understand by the end of this video." That last sentence is the most boldly incorrect assumption anyone has ever made.

[@og-ru6oc]

10/10 statement, imma rewatch it to try and prove him right 😂

[@Maxawa0851]

3 dimensional space, represented on a 2d screen trying to understand with your 1d brain

[@syedrafsanimam7406]

I kinda did understand it

[@mlmimichaellucasmontereyin6765]

Yep, even without admitting that a "4D sphere" is a purely illusory figment of maths, it's also misleading BS.

[@Maxawa0851]

@@mlmimichaellucasmontereyin6765 what

[@hamsterwheel16]

“It might feel weird to talk about two circles being perpendicular to each other, especially when they have the same center, the same radius, and they don’t touch each other at all…” I understand each word in that sentence individually, and that’s it.

[@DecemberGalaxy0]

maybe they existed in different times

[@killer_boss01]

I saw something like this on acid

[@user-zm5sk4ht9c]

Hogwarts geometry

[@sloweasy9273]

@@user-zm5sk4ht9c i thought this was an episode of the original outer limits

[@leodarkk]

They have to both exist at the same time at least once as they share the same origin.

[@entropyz5242]

"A line is just a circle that passes through the point at infinity" -- Grant Sanderson That is really deep, man.

[@taranloses]

god damn thanks for pointing that quote out so i can remember it. i just finished the video but my mind is still in the process of being blown.

[@rodschmidt8952]

And then there are an infinite number of such lines, going in all directions -- and for each direction there are an infinite number of parallel lines going in that direction. So, infinity squared.

[@strayedaway19]

1D infinite Sum -> 2D circular constant . So that is why PI pops up in linear infinite series

[@robinsonchukwu7295]

😂😂😂 I don't even get it

[@christianneisler1488]

Look up inversion about a circle @Entropy Z

[@paradox9265]

The animations are so intricate, the math must have taken you so long to put together and understand for yourself, let alone explain to any random KZbin browser. I express my deep gratitude for you taking the time to put together your videos, especially for free for our viewing.

[@yukiakihara2632]

indeed.

[@-danR]

I dunno I just empathized with Linus to such an extent that my head exploded. In all 2 directions.

[@LittleWhole]

I remember in my 5th grade classroom, there was a short fiction chapter book where the plot is based off of the main characters visiting 4D space. I think it was called "The Boy Who Reversed Himself" or something like that. I remember after reading that book I was absolutely captivated by the notion of higher dimensionality and went on gigantic Wikipedia rabbit holes trying to learn all about it. I never really totally understood it but this brought back memories - one of the chapters in the book talked about how the main characters were eating food in 4-space, but because they were 3D objects the 4D beings could see right through them in the 4th axis and were laughing their butts off at the food plopping down and going through the digestive system. It talked about how 3D retinas were required to see in 4-space and how a 4D object would cast a 3D shadow, and how a 4-sphere passing through 3D space would look like a random sphere popping into existence and expanding and shrinking out of existence similarly to how a 3D sphere passing through 2-space would be a circle doing that to a 2D observer.

[@mamatoshgupta]

- Carl Sagan

[@pixelprizm]

I read this book on my own in middle school and it was definitely a formative book. Got me thinking about the concepts of 4D space.

[@horaciomoreno2470]

You should read The Shape of Space by Jeffrey Weeks. Great introduction for a 3-d lander.

[@MysteriousSlip]

The fact that someone exists who can make an intelligent comment, but also was able to go on a Wikipedia rabbit hole in 5th grade makes me realize I am old...

[@LittleWhole]

@@MysteriousSlip You're not that old, don't worry. I'm only 17. I'm still in high school.

[@twigglesoft]

You know it's going to be a bumpy road when you don't understand the 1 dimensional analogy lol

[@ASLUHLUHCE]

Hahaha

[@unperrier5998]

you made my day

[@Guztav1337]

Practically impossible. Just go back and rewatch then.

[@ripinpepperonies9754]

shrdlu is that an insult to my 30 iq

[@Canyon_Lark]

@@Guztav1337 Friendly reminder not everyone's brain works the same way! While u and I eat this shit up and the spacial reasoning makes a lot of sense, other's may not have the same foundation, and put their brains to other things... and that's totally fine and dope. not very cool to tell someone it's "practically impossible" to not understand something. It's rude and false. It's super easy to assume that because something comes easy to you it must come easy to everyone but this is suuuuuper not true. learn on and be gentle.

[@yannisconstantinides7767]

22:00 "The number -1 is sitting off at the point of infinity, which you can easily find by walking in any direction." Ah, so simple.

[@SimonClarkstone]

Or flying off in any direction, which wasn't shown in the animation at the point.

[@KuraIthys]

Mmh. It's a projection of a hypersphere though, so in the actual geometry you'd have rotate your orientation into the 4th dimension somehow. I suppose what the video suggests is that you have a 4th dimensional being helping you out with that part. XD Really by analogy to what a sphere is like in 2d, where you can navigate the surface of the sphere as though it was a 2d object by treating what is actually a pair of polar coordinates pointing to the surface of a sphere as though they were cartesian coordinates on a plane (this is basically how we navigate the earth's surface after all). In a similar manner the 3 polar coordinates that specify your location on the 'surface' of a hypersphere could be reinterpreted as the cartesian coordinates of a volume. So a volume of space that is actually the surface of a hypersphere... Admittedly 4d geometry is near impossible to visualise, and even getting close causes big headaches. XD

[@patrickcoan5172]

That hit me too. If I'm lost, just walk to infinity, which is 1

[@fabianodauwe6811]

Maybe the best way to picture this is trying to represent the Earth in a map. Take the world map at UN's logo, for example. It's centered on North Pole, where is South Pole? It is actually the whole outer circle. The only difference is that to represent this idea correctly the Southern Hemisphere should tend towards infinity. North Pole dimensions would be underscaled, Equator would scale correctly, South Pole would be increasingly higher in scale as lattitude grows. New Zealand would look millions of times bigger than Siberia.

[@RedHairdo]

Infinity isn't a number, though, so that bit got me confused, but I believe I understood the idea.

[@evanwalker4672]

As a physics PhD student I thank you for helping me understand these things. Group theory is a whole lot simpler now

[@maxwellchiu9934]

I have watched this video at least 5 times in last couple years. I get better understanding every time. Kudos. Totally worth it. Your own work on producing this presentation is admirable.

[@patrickcoffey5933]

lol good ol' imaginary, jimaginary and kimaginary numbers

[@felixroux]

I can't even understand the 1imaginary numbers.

[@shin-mmxxiv-hna-official]

After... And limaginary And mimaginary And nimaginary And omaginary Before... And himaginary And gimaginary And fimaginary And emaginary And dimaginary IMAGINARIES LIMIT!!!

[@paramjitsarkar6414]

@@shin-mmxxiv-hna-official pimaginary qimaginary rimaginary simaginary timaginary vimaginary wimaginary ximaginary zimaginary

[@Farzriyaz]

type-of-number addition: real+imaginary=complex

[@mastershooter64]

lmaooo good one dude

[@johnchessant3012]

I think Lord Kelvin was onto something when he said quaternions are an "unmixed evil".

[@erik-ic3tp]

Why?

[@directordissy2858]

​@Ron Maimon What's the best book to learn real vector calculus

[@user-hq5fn6yv2v]

@Ron Maimon the thing is that the term vector as well as dot and cross products were discovered as part of the quaternion arithmetic. Then people just found out that vectors themselves are very useful. They generalized them to other dimensions, started representing linear maps with matrices, generalized mixed product to the concept of determinant and so on. Tensors came naturally when people stated studying polylinear maps. I don't see any way to explain tensors without first explaining vectors. The most natural way (IMHO) of defining a tensor is as an element of the tensor product of copies of some vector space and its dual space. There is also another way that is often taught to phisicists where tensors are explicitly defined by their coordinates, but that's much more confusing if you want to actually understand why they have such nice properties. It's like teaching matrices before linear maps. That's why quaternions are a "positive evil". There aren't the best way to talk about the 3D space as Hamilton thought, but without them linear algebra would've been discovered much later.

[@user-hq5fn6yv2v]

@Ron Maimon So your point is that actual linear algebra is more useful than 3D vector algebra with dot and cross products? I mean yes, that's definitely the case. It's just that while the cross product is ad hoc operation that only works for 3D vectors (unless you allow products of more than 2 vectors), the dot product is not. When I was initially introduced to it I was wondering how the hell it was discovered that by adding products of corresponding coordinates you get product of norms of vectors and the cosine of the angle between them. I understood the proof, but why would you even be trying to find a proof for that? The proof I was shown wasn't elegant or natural enough to act as a justification. It all become clear after I learned about quaternions. You are probably right about the general notion of a vector though. There are no products of two vectors in general vector spaces, and they probably got the name after Hamilton's vectors became popular.

[@user-hq5fn6yv2v]

@Ron Maimon well, the proof I know is pretty much the same, except the cosine part that was proved by considering the coordinate system specifically aligned to make computations easy. I guess we proved the law-of-cosines from that. The thing that I was struggling with back then is why would you even start with such a formula. In hindsight, guessing that dot product formula could be useful is much easier than guessing the rules of multiplication for quaternions. It's just that I cannot really imagine why you would be experimenting with nice looking formulae on vector coordinates, but the reason for experimenting with different multiplication tables on imaginary units was quite obvious.

[@Red-Brick-Dream]

Once again, this channel is in a league entirely of its own. The best of the best on maths-KZbin.

[@saireddy528]

My heartfelt thanks to you, 3Blue1Brown. No one could have done a better job. Thank you so much

[@arlenn729]

As an engineer, I pay all my respect to the people who deeply understand these complex math. These math have revolutionary effect on engineering productions these days

[@mastershooter64]

engineers: thank you very much mathematicians! physicists: you're welcome

[@Coolgiy67]

@@mastershooter64 lol physicists are really the back bone to everything

[@justrandomthings8158]

@@Coolgiy67 who tf gives physicists the math they need tho

[@Coolgiy67]

@@justrandomthings8158 other physicists 🤣

[@justrandomthings8158]

@@Coolgiy67 oh, the physicists make the math? You can use laughing emojis and stuff but it’s of course the mathematicians that do that, and then the physicists actually apply it. The job of a mathematician in todays age is much more noble, because no one will ever know their names, just the names of the engineers (not physicists either) that will use their work

[@acapellascience]

I almost always get an aha moment from your videos, but I think this is the first time I've gotten an aha moment on a topic I ACTUALLY COVERED already. Great job!

[@HerodotusVon]

acapellascience When he was talking about William Rowan Hamilton my brain automatically started playing the music in my head

[@Gravitraxer_AangCZ]

I also responded

[@TheViolaBuddy]

I actually only recognized the William Rowan Hamilton story about the bridge and his children from the A Capella Science song.

[@callmedeno]

I'm yet to get an aha moment :(

[@MultivectorAnalysis]

acapellascience, your W.R.H. song is absolutely brilliant! It's my favorite song on your channel, perhaps tied with your nerd manifesto, and just slightly ahead of Bohemian Gravity. This 3b1b presentation of quaternions is fascinating (I love it!), but it is only presenting their (historic) development as extending the complex number system. Have you looked into the Geometric Algebra (GA) representation of quaternions? GA provides a straightforward way to expand vector algebra into a multivector system that also manages to encompass complex numbers and quaternions in a completely natural way, capturing all of their computational advantages and overcoming their restrictions to 2D & 3D rotations. Complex numbers are very useful for rotations in the plane (R^2), and it turns out they're isomorphic to the even subalgebra of G(2) (the geometric algebra of R^2). Quaternions are very useful for rotations in space (R^3), and it turns out they're isomorphic to the even subalgebra of G(3) (the geometric algebra of R^3). There's a GA crash course on my channel demonstrating this beautiful unification, and I mention your William Rowan Hamilton song at one point too.

[@the_real_sublicious]

The background music is really relaxing and fitting, good job Vincent Rubinetti!

[@BenStoneking]

Wow. You deserve a nobel prize for the way you take such advanced and conceptually challenging topics and make them accessible to normal schmo's like me. Bravo and thank you!

[@CatzNGaz]

4D aliens somewhere are using this video as a reference to better explain 6D spaces, ...

[@Joyexer]

We also have Sedenions. Those are 16 dimensional... :)

[@MrRyanroberson1]

@@Joyexer sedenions are for 5D space. 1d = 2^(1-1), 2d = 2^(2-1), 3d = 2^(3-1). 4d is octonions, 5d seden, and 6d is that 32-nion system. But consider: octonions are nonassociative, so any physics in 4d space HAVE to involve alternative processes, or simply can't happen.

[@Joyexer]

​@@MrRyanroberson1 Hmm so you say, that 16 algebraic dimensions are not indicating a 16 dimensional space? I m not actually studying this subject...

[@SmartassEyebrows]

@@MrRyanroberson1 That doesn't seem to be the case? Octonions are an alternative form of algebra that is "8D" in algebraic reality/space (note that this is not the same as physical reality/space). It has important uses in Special Relativity, but they are not a physics description of 4D space. 4D space wouldn't have any special physics compared to 3D, just more "space", as now everything lies in 4D points. Quaternions are a 4D number system, but that is unrelated to 4D space, just as octonions are unrelated to 4D or nD space. They are extensions of or fundamentally alternate number line realities, respectively. Octonions are interesting since they have power associativity and most interestingly, subalgebra associativity (any 2 elements are associative, but not the whole). And of course, sedenions are an extension of octonions (like quaternions are an extension of the complex number line). Crazy stuff.

[@MrRyanroberson1]

In order to correctly represent a transformation in N+1 dimensions, you need 2^N many axes. One of them is real, the rest imaginary. Quaternions = 3 im + 1 re = 4 = 2^2 -> 2+1 dimensions. The Nions are *algebraically* N dimensions, but represent log2(N) +1 spatial dimensions.

[@smartereveryday]

{Pauses Video} A couple of things. 1. Holy Crap. Grant is about to explain quaternions? (grabs popcorn). 2. Oliver Heaviside @1:57 is freaking Wolverine / Hugh Jackman. That is all {Clicks Play}

[@demetriosavdalis5574]

Both your channels make us smarter every Day. Really. Good vids the both of you

[@lordkelvin100thompson8]

If you read Hamilton's original paper, it's obvious he knew *exactly* what he had developed. The clue is in the very name: qua(d)-ternion -- a 4-D entity that consists of a scalar plus a vector. In 1843 few people were ready to accept a four space with time as one of its dimensions. That's why they resisted Hamilton all the way until the Gibbs-Heaviside Simplification allowed everyone to breathe a sigh of relief that they wouldn't have to think about quaternions and Hamilton's four space any more.

[@keylanoslokj1806]

important upcoming particles: 1)asspullions 2)quackernions 3)unicornions 4)cluelessinos 5)badsciencions

[@ankittiwari1647]

Hey dastin here! from smarter every day say hello to him we are also a big fan of your channel.

[@izuix5629]

3. Peter Tait @1:51 is Sam Gorski

[@michelcazayoux2334]

Rewatching this (again). I first found this from you years ago. I think it is SO beautifully done. You have a gift - and the tools to share it. Thank you!

[@yashsharma6112]

I really cried after watching such a beautiful explanation out of happiness. Hats off to you & I finally understand how to see the beauty inside mathematics. You did a great great great job. Thank you

[@CuulX]

I've waited 10 years for this video. Whatever time you spent on this video was justified! Wish I had money so I could become a patreon. This is the best channel on youtube!

[@3blue1brown]

The whole point of Patreon (well, in my mind at least) is that content which is better funded by a direct consumer-creator relationship than by a primarily ad-driven model can still remain free for those with little-to-no disposable income. So if you find the content valuable, just sit back and enjoy, and feel glad that the system is working.

[@Yous0147]

That is such a fantastic way of thinking about your content.

[@unperrier5998]

10 years in how many dimensions?

[@mastershooter64]

@@unperrier5998 10 + 1

[@mlmimichaellucasmontereyin6765]

@@3blue1brown > Thanks for that reminder. My pages @ Patreon will soon feature my preprints, etc., some of which will reference your videos, etc. Kudos ~

[@ahmedal-shabi6032]

"one you will understand by the end of this video" Wow, some confident statement right there buddy 😂😂

[@GrahamMilkdrop]

There's no getting away from the fact that it is a crazy mind-bending concept that most people will never need to understand!

[@joshuadsims]

Basically, when you multiply two three dimensional vectors together you get 4 different numbers. A Real number, and three vectors in the direction of i,j, and k: (ai+bj+ck)(di+ej+fk) = a + bi + cj + dk. In four dimensions this just rotates and transforms two points, rotating a point along a circle perpendicular to a point on the unit vector axes that is being multiplied. This plane is created by intersecting a unit sphere, where the real part is the z axis, and then plating the line that it intersects with on the plane through the middle of the sphere from -1. kzbin.info/www/bejne/mmWomJWdidJjeMkm47s Showing that the complex plane is basically a projection of the sphere onto a flat surface through a fixed point on the z-axis, which is -1. From this idea, we take the unit circle on the complex plane by wrapping it around the equator of the sphere being projected onto the plane. We then rotate the sphere and this has the effect of rotating and transforming the points relative to the angle at which we rotated. So let's say we multiply two quaternions together, this gives us some point a + bi + cj + dk on the unit sphere. The a, or height on the line through the z-axis of the unit sphere projected onto the complex plane, is the transformation based on the angle by which the i, j, and k axes were rotated. For example, if you rotate by 90 degrees on the ij unit circle you will rotate the i1 axis by 90 degrees, and thus the jk circle around the i1 axis by 90 degrees, and move the point from 1 to i along the i axis, which is equivalent to a 90 degree rotation on the i1 unit circle. Basically you have 3 circles wrapped around eachother at 90 degrees and the number 1 in the center. The 3 circles are the ij, jk, and ik unit circles. These form a sphere with 1 at the center. Rotations around these unit circles correlate to a rotation on the unit circle which corresponds to the unit circle created by the unused 3rd complex number and 1, which is projected as being just a line through the center of the rotated unit circle, which is seen as a transformation of a point equal to the rotation angle on the unit circle. For another example, let's say you rotated the jk axis by 90 degrees, this is equal to a 90 degree rotation on the i1 unit circle, which, because it is projected onto a straight line through the point -1, results in a transformation to the point i from 1. kzbin.info/www/bejne/mmWomJWdidJjeMkm21s Hope that makes a little more sense. Essentially rotations of one unit circle rotate another unit circle, and when a point is projected from these unit circles onto a plane through the center of the hypersphere that they create they form the complex plane that we can see in 2D or the stretching of the line intersecting this projected plane as a stretching to and from infinity in 1D.

[@ahmedal-shabi6032]

@@joshuadsims thanks mate, but I already understand it I was just "trying" to be funny. I hope people profit from your explanation

[@Relaxing137]

@@joshuadsims (ai+bj+ck)(di+ej+fk) = a + bi + cj + dk. So its like just a decomposition of the what happen when a certain coord change from point a to point b on a sphere surface using linear translation of axis, is that it ? trying to figure out, also I think I kinda maybe understood that the scale of distance on the 3d representation isnt visualy representative for the sake of representation but doesnt really matter on application right ? (ai) is like a triangle to get to the point on the offset a, still I can hardly understand why there's so much variable when I'm used to see x,y,z xD. I think I understand the concept, but I really can't relate the formula to what I see in my head, I guess that's where practice and theory differ ¯\_(ツ)_/¯

[@joshuadsims]

Relaxing Gaming Basically all rotations transform x, y, z points. This just represents the transformation through 4 vectors rather than 3.

[@AhmedOmar-uf3gu]

Besides the absolute fact that I can truly understand a concept after watching your videos, I am also greatly inspired by the thought of the amount of effort and time you put into your work. Thank you for all the quality and benefit you provide, and this beautiful art that you create.

[@spoingus20]

this is beyond amazing, the amount of work you must have put understanding it yourself and then all the amazing animation in the library you coded yourself 😍😵‍💫

[@feynstein1004]

Brain.exe has stopped working

[@pi5549]

Try migrating to a UNIX based brain-OS

[@feynstein1004]

+pi Ahaha I'll try, mate. But even UNIX based OS will have executable files, no? :P

[@Thanatos2996]

@@feynstein1004 yes, but no self-respecting *nix operating system will use .exe as an extension. *nix has the executable bit, no extension needed.

[@bigbox8992]

Apt-get upgrade

[@iisthphir]

You need to download intelligence.dll and put it in the system directory or enable autoupdates through the ear hole by enabling the notifications for this channel. It may demand installation of some dependencies first though if you do it that way.

[@MM0x2A]

You are bonkers, my dude. As someone else pointed out, this might actually become _the_ standard reference on quaternions, which is quite amazing. Also, its an absolute pleasure to listen to that voice. Thank you!

[@repositorytutorial3d50]

You are my hero man, I'm not a mathematician, but I needed to use quaternions for a research I'm doing at university, I didn't want to apply them blindly, I really wanted to understand them, but no source on the internet was enoughly simple for me to understand. Then I found your video. Your ability to visualize complex problems is remarkable. I can't thank you enough. if you come to italy please let me know I will offer you a dinner.

[@jethrolarson]

"A line is just a circle that passes through the point at infinity" *head explodes*

[@CathrineMacNiel]

well, there are number circles.

[@VinayKumar-vu3en]

Legends say that Linus is still figuring out what Grant taught him.

[@pizzarella985]

i'm kinda in the same boat with Linus there it's kinda rough sailing a one-dimensional boat but that's what i get for being a dumdum

[@maxwellsequation4887]

And Felix is already dead

[@heavennoes]

@@maxwellsequation4887 I see you in every 3b1b video even in lockdown math.

[@beroyaberoya6856]

While drinking from his LTT water bottle

[@PaulPaulPaulson]

This is going to become THE standard reference for learning quaternions!

[@thedutchflamingo9973]

I sure hope so

[@Chainerlt]

+1

[@josephgroves3176]

Now just to unify and standardise the notation, and we can put quarternions back into vector calculus...

[@zairaner1489]

No

[@abhimanyupallavisudhir6007]

No, it won't. To be honest, this wasn't a great video -- no real motivation was introduced for having a pair of rotations for each quaternion, and the reference to the stereographic projection, while nice, doesn't really introduce any new ideas either. The correct motivation for understanding quarternions comes from tensors and bivectors, or alternatively from trying to generalise complex algebra to three dimensions.

[@onemightsay248]

Just want to appreciate how much thought goes in to the animation such that it makes understanding the topic that much clearer.

[@AleksanderFimreite]

If I'm not mistaken, this helped me understand how scaling the "Field of View" in 3D applications works. That is essentially scaling one of the axis which is pointing horizontally in relation to the camera if I understood this right. It still does not help me one bit to grasp which modifications I have to put into a quaternion to reach the exact point I want. But luckily, applications tend to provide useful features to dumb it down to a matter of providing relative angles.

[@Jay-kb7if]

I'm getting that sense that people often think global space and not local space and makes it harder to grasp.

[@thatmusicalguy7]

"One which you'll understand by the end of this video." ...right

[@Anvilshock]

Confirmed, that was a bold claim.

[@tanner1985]

@@Anvilshock I always feel dumb and I always postpone understanding in a later time. When this later times will be, I dunno.

[@Wertsir]

@@tanner1985 Sounds like a problem for future me. That guy’s a sucker.

[@rlenclub]

lord kelvin: quaternions are evil game programmers: damn straight

[@asandax6]

And that's why I let Unity do the Calculations While I just use the Quaternion Method. 🙂

[@Kaldrin]

Yes I'll stick to EulerAngles it's already complex enough

[@matejlieskovsky9625]

@@Kaldrin No, no, no! You don't understand! Once you break your mind badly enough to see the intuition behind quaternions, quaternions will *be* the simpler method! :-D (I'm studying theoretical computer science. I am almost convinced the main benefit of analysis in the first year was us getting used to gluing our brains back together.)

[@Kaldrin]

@@matejlieskovsky9625 STAND BACK SATAN

[@boomerboxer3574]

@@Kaldrin but to be honest, euler angles are prone to gimbal locking if you are not careful.

[@Rationalific]

This is some brilliant stuff! By the way, the way I think of a 4D tesseract is to think of it as a 3D cube moving in time (the 4th dimension) or increasing or decreasing in size. The tesseract illustrates this movement at a single point in time. This illustration in the video also appears to be a sphere increasing to an infinite size and back (or being zoomed in on infinitely and back). I just thought I'd mention that.

[@zarblitz]

Watching this video is the first time I've been able to understand 4D object as a projection onto 3D space. Showing each dimension as a projection on lower dimensions really helped me visualize how 4D is projected (and why we can't really comprehend it directly).

[@pharaohjohnson7627]

This has to be the best explanation of "imaginary numbers" I have ever encountered in my life. I have never been good at grasping certain math formulas until I see how and why they were invented and what they apply to. Basically, show me the mechanics of it and I'll be able to use see it in my head and use it in practical and theoretical applications.

[@rummanadib217]

Same and I think it's basically how Mathematics should be learned

[@necroowl3953]

This does not explain imaginary numbers, this positions imaginary numbers in the context of rotating planes Didn't cover x = sqrt(-1), which truly explains what imaginary numbers are. This is one of their applications. It's a system

[@pharaohjohnson7627]

@@necroowl3953 It does explain "i" in more detail for me.

[@googa8859]

Your animation just blows my mind every single time. These visualizations should be used in all schools

[@QAYWSXEDCCXYDSAEWQ]

The explanations and graphics on this channel are world-class, really excellent!!

[@jessfucket]

i watched this twice and the last part 4 times. you explain things like feynman does. you capture the essential simplicity.

[@ASLUHLUHCE]

After watching a fascinating video on visualising quarternions, I should get back to my school work of adding and subtracting vectors lol

[@ntwede]

Yeah, but do you know how to multiply them as well?

[@ASLUHLUHCE]

Pfft. I've watched through 3b1b's amazing linear algebra series

[@BlindManBert]

Quaternions can be thought of a specific application of linear algebra, or as a generalization of a real number in the same way that complex numbers can be thought of in terns of linear algebra. Ultimately, I would think that this overlap of interpretation comes from abstract algebra: the group theory associated with complex numbers or quaternions over the addition, subtraction, or multiplication operators. In linear algebra terms, quaternions can be represented either by 4x4 matrices of real numbers, or 2x2 matrices of complex numbers. Wikipedia has a nice section describing this equivalence.[1] [1]: en.wikipedia.org/wiki/Quaternion#Matrix_representations

[@Psyle_]

@@BlindManBert Yes in one sentence, there is a group isomorphism between the quaternionic sphere S and SU(2) Or to bring it back to common rotations S/{+-1} ~ SO(3) Lol that's actually the school homework i gotta get back to.

[@angelmendez-rivera351]

Blind Man Bert Another formulation that is also equivalent is to treat quaternions as a special class of paravectors, and the quaternions product is equivalent to a sum of the dot product and the cross product in Euclidean space for 3D vectors.

[@natejenkins9700]

"... one which you'll understand by the end of this video." A little optimistic, aren't we?

[@shingsey8349]

I've slacked off watching this video because it wasn't the time I could concentrate and learn about quaternions. This is the first video I end without having a real grasp on what happened. Not because of explanations or illustrations, on point as always, but just because it is a too complex subject to get an intuition with one video. I feel a bit defeated, as a pretty good math student xD I'll come back to it and understand it later, i promise. Great video 3B1B

[@andrewmaperson]

I also find that rewatching these vids a few days later helps:) what grade are you in, im in high school?

[@angeldude101]

Quaternions have 4 components, but they themselves have nothing to do with 4D space. Quaternions are fundamentally 3D, so I actually consider this video one of the weakest of 3B1B's. Quaternions are only treated as 4D because complex numbers are mistakenly treated like ordinary 2D arrows, which they're not. Complex numbers are just as much axis-angle as quaternions, only with only 1 available axis, so it only has 1 imaginary component; that axis being the origin. (The point, not a line sticking out of the page.) The scalar part says nothing more than how much _not_ to rotate. It's the identity, both in terms of multiplication, and transformations. Every quaternion and complex number is a linear combination of a 90° (or 180° because of quaternions' double cover) rotation and a 0° rotation.

[@dan5801]

@@angeldude101 Thanks! Could you please suggest some more intuitive tutorial for understanding Quaterion?

[@angeldude101]

@@dan5801 Probably my favourite would be "How to think about Quaternions without your brain exploding" from Alex Rose I'll link it in a separate comment to make this comment less likely to get eaten.

[@dan5801]

@@angeldude101 Thank you so much for prompt reply! I will definitely watch it! 😀

[@TheAIEpiphany]

This kind of education coupled with VR or MR/AR/XR (whatever your favorite abbreviation is) is the future of teaching. We need to exploit the fact that the human visual system is powerful and the fact that our spatial memory is even more powerful. Amazing work Grant! You're reinventing the way mathematics is presented.

[@Tremuska]

VR is the pure evil to teach something

[@mlmimichaellucasmontereyin6765]

Of course, so is Marty & the Mathologer crew (and the Numberphile crew).

[@motherisape]

It's not educational video it's just fun video to enjoy . you can't explain the formula without traditional way .

[@Gr3ypsTFT]

@@motherisape But this video did explain the formula.

[@motherisape]

@@Gr3ypsTFT nope you need to derivative the formula you should know how it works

[@arvasukulkarni3686]

Grant: "It may feel weird to have two circles perpendicular to each other," Me: No, it feels normal. Grant: "especially when they have the same center, radius and orientation" Me: *mind explodes*

[@MrCubFan415]

Arvasu Kulkarni 🤯

[@adityashankar5267]

🤯🤯🤯🤯🤯

[@jeupater1429]

Well actually we know exactly what this is. if you know where to look. The relationship between subjective consciousness and the objective conception of the world is exactly so. From a subjective perspective, it feels as though objective space is entirely contained in the subjective. While from the objective perspective it seems as though the subjective mind is entirely contained in objective space. So two interlocking spheres with the same center is exactly what we are. Cheers

[@orhunyeldan9800]

think one of the circles in the past and the other one in the future. so our 4th dimension is time.

[@joaopedro-mm5lf]

@@orhunyeldan9800 time? that's not how dimensions work

[@WyroX15]

Wow, that 30 minutes passed so quickly... such a captivating and beatiful video

[@claudiodipietro9489]

Flipping amazing ! Thanks for the videos you post and sharing your incredibly intuitive approach to extremely complicated math

[@luchoPueyo]

The graphics for this video are just stunning. Amazing work!

[@timedilatesme]

One of the best visuals someone can ever give

[@thenotflatearth2714]

The one has awakened again

[@jannisopel]

Weird I just read Call of Cthulhu then I see this comment.

[@ArcAngle111]

Nah man be careful lol

[@arturocrespo3699]

The Spherical Earth awaken my masters ayayayayaya

[@cristicristi2668]

no, it's past noon

[@ArcAngle111]

You mean the zero? 😉

[@RamsesYT]

This is a great explanation of quaternions, progressing through dimensions us humans naturally relate to, into the unfamiliar 4D world that, in so doing, you seem to have made me feel familiar with - made me expect it, even. Thanks so much for your wonderful lessonsl! 😊

[@mlmimichaellucasmontereyin6765]

RYT - Cool, yet remember, the 4th "dimension" is a virtual conceptual construct of maths, for the sake of convenience only. Anybody who thinks otherwise fails to understand the logic and metalogical principles that enable maths & geometry (etc.).

[@ILiberoI]

For everyone who just saw the video, I can really recommend the interactive lesson, which are recommended at the end. They are are really well made and help understand the topic even further.

[@potatooflife8603]

For those who are having trouble getting it, rewind! Rewind, then rewind again. And again. And again. Eventually all the pieces begin to fall into place. But don't forget to be kind and patient to yourself until you get there.

[@rafaelgomez4763]

Potato of Life kindness does go a long way.

[@stancartmankenny]

Be kind - rewind.

[@davidlightfoot3217]

I'm 15 and I'm watching this. I find that kind of weird but oh well.

[@avi12]

Although it's still unclear, the way you attempt to make me understand this topic using these animations, fascinates me That's what I love about this channel - math + animations. The best way to understand math is by visualizing it

[@victoriab3308]

The animations are a total game-changer in the comprehensibility of this stuff!

[@TheRockybulwinkle]

Seeing that scale and rotation of complex number multiplication solidified something in me that my years at college could not, even when working with polar coordinates where the scale and rotation is practically spelled out as much as it can be without an animation.

[@JoelDowdell]

I said this on the 2^20 Q&A video: "If you could put a good explanation of quaternions out there that would solidly put you as the best math content creator out there in my mind. (Yes even better than vihart)" This obviously had a lot more preparation than one week, and he mentioned the idea in the video so he was already thinking of it. But still, I said that and one week later this video comes out.

[@d0themath284]

> This obviously had a lot more preparation than one week Or, he's just super competitive and really wants to be "the best math creator [ever]."

[@tonee899]

Who's vihart

[@columbus8myhw]

tonee899 Look her up, she's good

[@roygalaasen]

tonee899 you should look her up! Her videos are worth watching even if she is a tau supporter!

[@brunohentschel3023]

Tau? It's treason then. /s

[@superj1e2z6]

Animations are so superb No, They are beyond superb. Math's so pretty to look at.

[@roberthuber2770]

Quaternions... by far the spookiest mathematical object I've encountered. Chirality and rotation deeply built into the system. Incredible!

[@MiScusi69]

This was the most entertaining video ever. I literally watched for the entire half an hour, and it felt like 10 minutes!

[@spikku]

I wish I could suddenly realize things like that on bridges...

[@anithaajaien6391]

Explaining 4D objects using 3D shapes in a two dimensional screen to my 1D brain 😁 Edited :- After a year when I'm back, : what's happening here 😳😂

[@veganath]

Comment of the day award!! Thumbs up

[@lostmontanelas4236]

Good joke, exactly like one i heard on that A.I comedy channel explaining dimensions...

[@RenderingUser]

mine's 0D

[@simopelle]

g(old)

[@mbenitez6722]

OOoOooooooo

[@aggelossirbu7431]

The scalar and vector parts interpretation of quaternions reminds me a lot of the electromagnetic potential four vector! This is so facinating!!😃😊

[@educationonline2485]

Possibly the best explanation I have heard for anything this complicated. Thank you !

[@UniquePerspective]

13:39 - How to confuse both flat-earthers and sphere-earthers

[@vari1535]

BAHAHAHAHAHAAH

[@Guztav1337]

what?

[@danboekenoogen4957]

Wake up America, the earth is a sphere projected on a plan.

[@goyonman9655]

😂😂😂😂😂

[@jorritwoudsma7530]

Its such a beutifull animation

[@FreshBeatles]

when you portray the audience or students as the pi symbol, are you saying that they are irrational students?

[@christopherbrooks3691]

transcendental, my friend.

[@1e1001]

Linus the linelander is in the shape of an i, which makes sense, because he is imaginary

[@shanineedwards6894]

Lol

[@guyarbel2387]

Lol

[@Taurui]

You are what you eat :)

[@NegInfinity]

This is extremely useful. I've tried many different explanations, this is the only one that allowed me to understand it intuitively. I think it would be even more intuitive if there was a VR application visualizing this. Because in a VR helmet you can see it in 3d.

[@intrifix4536]

This video is an absolute gem. Thank you for making this.

[@mastershooter64]

25:45 that 4D creature is known as Grant Sanderson

[@EidosGaming]

Like the Fourier Transform, or Riemann Zeta even if not completely, this is the kind of thing I thought I would never understand, until you make a video on it

[@shreyasraut6224]

that is so aptly said.

[@NoahHornberger]

around 27 minutes I got an intuitive glimpse of the 4D hypersphere. It was well worth the investment. After years of just not being able to wrap my head around it . . . . thanks!

[@GumbyJumpOff]

Damn. I'm jealous.

[@noepopkiewicz901]

Great video! This concept is counter-intuitive, but it's logical and follows a set of rules, there's a pattern it always follows and that can be written down as a formula. The amount of effort and thought that went into this video is incredible. The 1D and 2D analogies really help to grasp this concept. Playing around with the interactive visuals site is also a massive aid.

[@garrettstambaugh6271]

I've only just discovered this video, and it's honestly amazing. In trying to wrap my head around 4d maths, it's always either explained as purely mathematical or visually with the same tried, true, and not tremendously helpful "shadows" or line extensions. Although the analogy is something I have encountered before, this model is not something I've ever come across anywhere else. And it's tremendously helpful, at least for me, in visualizing something that is otherwise impossible to visualize. So thank you. It's been a big help!

[@PuzzledRandO]

17:49 I see what you did there. 3.14 159 265 358 979 323 846 264 338

[@dsdsspp7130]

oh...

[@ceskale]

Great catch

[@michaelwang1730]

@@memefief8527 where?

[@MrManojKumarPrasad]

Ohkk good catch

[@coopergates9680]

(Golden ratio weeps in offense)

[@fders938]

I've rewatched this video from start to finish 4 or 5 times now, and it's finally starting to make sense

[@noormili3230]

This is brilliant, you are brilliant, thank you for translating such abstract concepts into concrete visual animations. Doing math is great, but SEEING math is amazing.

[@pendalink]

ive been trying to imagine those 3d projections for years and it feels SO damn good to see them animated and explained like this, thank you so much

[@NoobMasterX1]

The quality of this channel is supreme!!!!!!!!! Love this video very much

[@antonyguglielmone5430]

Super tight. Thanks for tanking the time to create this. I went through my classes with nothing but my own intuition on such things. Surprisingly I wasn’t horrifically far off 💃 Insanely helpful to see it broken down this way.

[@andrewthebladethrower8371]

amazing! the visualization is remarkable and very intuitive. I have been building my knowledge base of complex numbers for over a week now, this lecture is outstanding. Especially rotation and projection parts, help in understanding on intuitive level.

[@qwadratix]

Even back in the 60s I was taught this stuff instead of the more usual vector approach we use these days. Back then, math in physics was really, really hard because we didn't know how to use it efficiently or smoothly. Imagine trying to understand curvilinear geometry using these things.

[@GnomeCore]

"One which you'll understand by the end of this video" haha ya if i weren't dumb

[@AdminAnish]

Haha😂😂😂

[@santiagosanz4157]

Dude you trolled 3b1b so hard dude!

[@Internetzspacezshipz]

I literally cannot even into quaternions.

[@jackdulin8242]

It takes more than watching one video to understand quaternions because learning math in general is a long and arduous process. You would have to read some mathematical literature, absorb the information, maybe get stuck somewhere, hit a wall, watch other videos related to what you don’t know, perhaps go on Math Stack Exchange, sleep on it for several days, maybe have a flash of insight after wrestling with the concept for days on end, keep reading, do some problems, get stuck on those, etc. Learning math isn’t easy, but it’s not an affront to your intelligence. I see it as climbing a ladder, you need to take it one step at a time. Learning math is hard for everyone, even the most intelligent people in the world. For example, it took 7 years for Andrew Wiles to prove Fermat’s Last Theorem. An error was found in the original proof. He almost gave up, but about a year later he submitted a correct proof. Anything worth doing will require sacrifice.

[@Autists-Guide]

Hehehe. Yup. Linus was light-years ahead of me on this one.

[@Jasomniac]

I dont think there is another youtuber that can explain abstract maths as good as you. You are trully the youtube GOAT in maths explanations

[@BesTheDemon]

Love the structure of the video + soothing music well adjusted to your voice. Other channels could learn from you :D. Rewatching two times was worth the knowledge. Keep up the work!

[@ytsas45488]

Holy shit, yes! I've been waiting for this for so long. Will probably be amazing, as always. Edit: Extremely intuitive and elucidating. Also, your 3D animation is beautiful! It's leagues beyond the 3D you used to make with conical arrows and such. This is ridiculously impressive. And, as always, the best sound design is unnoticeable until you look for it. Kudos to Rubinetti for that as well. Can't wait for that conjugation video!

[@kurumi394]

Me at 2230: alright I have a busy week in front of me so I need to sleep now Me at 0300: hehe yes qatrnons Edit: the way I express time has caused a small war in the comment section oops

[@--.._]

bruh i thought u were talking about years

[@martinhawes5647]

@Zero Six Most people express time that way. Most people in Europe don't even know what conventions are used in the miltary.

[@cookiequeen5430]

corona lockdown has its bright sides. no busy weeks for me soon, yay...

[@selfmodded9736]

wow it's exactly 3am on the dot. i'm done

[@MrNerd-ll9my]

quortrornons

[@macronomicus]

Love this vid, its helped me grasp the concept better than any other format ive seen, being a visual thinker. I've watched this a few times in the past, but only just now noticed you mention conjugation near the end. I think you might enjoy exploring "Phase Conjugation" coined by engineer Dan Winter of Implosion group, its quite intriguing, & a fundamental key of a great many things too numerous to mention here.

[@TheDeepDiveLLC]

You described this in a way where I think I was physically capable if visualizing it. It's hard to describe how I perceived it, just like it's hard to explain 3d to a flat lander, but I believe you allowed me to grasp that knowledge through the explanation of Line-us that's never been described to me before. Anyways, thank you.

[@arlwiss5110]

youtube: hey how about your screen becomes a 2d projection of a 3d projection of a 4d plane me: me: me: me: go on

[@kashthetrashman]

Googly Eye I

[@Yous0147]

This was honestly incredible. Thank you so much for sharing this with us. I tried to understand as much as I could, and you did a great job in visualizing and contextualizing something so complex so it made sense even for someone who isn't well versed in the math. It's honestly mind blowing.

[@cameliac9620]

this made me think of all those cool pictures of black holes we've been seeing lately. good job bro! I love your videos! mesmerizing!

[@kristinaswanson6063]

I wish you could have been my maths professor when I went to university. Exceptional presentation and clear graphics. Well done !

[@AlucardNoir]

The Rubik's cube analogy was quite inspired.

[@chipgruver2911]

My mind is blown. For a couple of moments, I almost saw into 4D space. This video is wonderful. Thank you for putting the effort into it.

[@thinkingAutomata]

Thank you for such a beautiful explanation and illustration.

[@realbartonjames]

This is an INCREDIBLE representation. Thanks!