Visualizing quaternions (4d numbers) with stereographic projection 

Underrated comment

You win this comment section.

*Cries in octonions*

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

brilliant!

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

Spoiler Alert!

Next time: 9 Imaginary 0 Real

Niooocce!

*dies*

I barely understood the 1 dimension

​@@MarloTheBlueberrySame

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

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

I kinda did understand it

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

@@mlmimichaellucasmontereyin6765 what

I can't even understand the 1imaginary numbers.

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

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

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

lmaooo good one dude

Hahaha

you made my day

Practically impossible. Just go back and rewatch then.

shrdlu is that an insult to my 30 iq

@@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.

indeed.

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

agreéd

- Carl Sagan

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.

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

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...

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

maybe they existed in different times

I saw something like this on acid

Hogwarts geometry

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

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

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.

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.

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

😂😂😂 I don't even get it

Look up inversion about a circle @Entropy Z

well, there are number circles.

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

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

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

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.

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

Why?

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

@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.

@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.

@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.

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

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

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

@@justrandomthings8158 other physicists 🤣

@@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

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

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. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-d4EgbgTm0Bg.htmlm47s 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. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-d4EgbgTm0Bg.htmlm21s 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.

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

@@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)_/¯

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

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

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.

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

@@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.

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

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

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

@@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.

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

@@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.

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.

I sure hope so

+1

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

No

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.

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

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

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

@@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.

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.

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.

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

10 years in how many dimensions?

@@unperrier5998 10 + 1

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

Damn. I'm jealous.

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

And Felix is already dead

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

While drinking from his LTT water bottle

transcendental, my friend.

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

Lol

Lol

You are what you eat :)

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.).

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

I also responded

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

I'm yet to get an aha moment :(

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.

Try migrating to a UNIX based brain-OS

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

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

Apt-get upgrade

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.

Potato of Life kindness does go a long way.

Be kind - rewind.

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

Confirmed, that was a bold claim.

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

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

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

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

@@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.)

@@matejlieskovsky9625 STAND BACK SATAN

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

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

Nah man be careful lol

The Spherical Earth awaken my masters ayayayayaya

no, it's past noon

You mean the zero? 😉

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

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

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

Arvasu Kulkarni 🤯

🤯🤯🤯🤯🤯

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

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

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

BAHAHAHAHAHAAH

what?

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

😂😂😂😂😂

Its such a beutifull animation

oh...

Great catch

@@memefief8527 where?

Ohkk good catch

(Golden ratio weeps in offense)

yes, yes and yes. you have plenty of company! ; 0

> 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]."

Who's vihart

tonee899 Look her up, she's good

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

Tau? It's treason then. /s

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

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.

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

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

3. Peter Tait @1:51 is Sam Gorski

Haha😂😂😂

Dude you trolled 3b1b so hard dude!

I literally cannot even into quaternions.

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.

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

that is so aptly said.

Comment of the day award!! Thumbs up

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

mine's 0D

g(old)

OOoOooooooo

Who knows, maybe the half video he talked about will be stuff exactly like that

Quack quack 🦆🐣

Repeat and practise if you need to actually understand and work with it :) I found excercises run.usc.edu/cs520-s12/quaternions/quaternions-cs520.pdf, under "quaternions"

VR is the pure evil to teach something

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

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

@@motherisape But this video did explain the formula.

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

3imaginary1real.com

@@3blue1brown Sadly, the site is under construction for a totally awesome idea still being worked on. pepesad

@@3blue1brown This site really exists? I was thinking some guy in the comment section was using made up word lol

Googly Eye I

Your bachelor sciences degree?

@@rohanshetty2937got 1 more year left at uni... yes, BS in CS to be exact... currently struggling with it because my maths background is not that strong

It’s even more beautiful when you do understand it. Even regular complex numbers are mind blowing, but quaternions are next level awesome.

Me too 🤯🤯

BS is just BS by itself, APSC is learning the applications of BS.

I understand you.

I'm watching this for the 3rd time and I'm slowly getting it.

I can't comprehend how people understand this lol