Quaternions

[@lopezb]

Beautiful lecture, thanks! Just the right amount of detail. Quaternions were invented by William Rowan Hamilton (also invented Hamiltonian Mechanics) in 1843. Heisenberg was one of the fathers of Quantum Mechanics in 1925.

[@KaiseruSoze]

I was going to point this out too. But I was betting someone else spotted the error. TY.

[@joaogonzalez4082]

Yep, I was going to state that also. But Gibbs did simplified its math to vector algebra as we know today 😏

[@gokceyildirim8161]

Heisenberg might have invented octonions to explain particle spins for quantum mechanics

[@DellHell1]

He said Heisenberg because he wasn't certain who it was. But when he stood still he became certain it was Hamilton.

[@takshashila2995]

Uncertainity principle.

[@gavtriple9]

Takshashila underrated comment

[@AlfredEssa]

Hamilton, not Heisenberg.

[@random_guy6608]

Idiot hamilton thinking about quaternions on his way to Party

[@robrick9361]

I heard Hamilton used his knowledge of Quaternions to become a drug kingpin. I AM THE ONE WHO EXTENDS COMPLEX NUMBERS!

[@JimAllen-Persona]

Guess he was uncertain 😂. Another Newtonian or Gaussian type legend (Gauss’ solution to the parallel postulate). As bad as that joke is, this is my first exposure to these... very interesting.

[@abenedict85]

@@random_guy6608 show some respect for your intellectual masters

[@That_One_Guy...]

So that's why electrons location are uncertain, because they're 4d beings

[@JackLe1127]

best part about watching youtube lectures is that you gain the knowledge but you don't have to do the homework

[@karz12]

You can't gain the knowledge without doing the homework.

[@ZeusLT]

why not

[@johnjackson9767]

+karz12 Word.

[@s.u.5285]

i prefer saying..best thing about you-tube college learning is you gain the knowledge without having to pay for it.

[@That_One_Guy...]

Advantage of online learning : 1.Gain knowledge 2. Choose to do or not to do homework (with freedom to choose when to do one) 3.Sometimes a much clearer explanation than your lecturer tried way too hard to explain (for math i loved this so much) 4. Need just a waaay shorter time time than the boring and weekly long explained things in your college 5. Free of cost 6.Never get left behind because of the bullshit limited amount time (see point 4) 7. Learning becoming much effective also because you're free from stressfull environment (annoying and noisy idiot kids who keeps babbling about something trivial, bullies) (i feel like stressful environment is one of the biggest obstacle of studying properly beside worst teaching and limited time BS) Why does offline learning isn't removed yet sigh. For anyone complaining about social interaction for same age, i ask you how does people in the past (where school isnt even exist yet) interact with each other ?

[@LibrawLou]

Excellent introduction via rotations, but the discoverer was Hamilton, not Heisenberg.

[@LibrawLou]

Pharap Sama History otta' at least be in the right century...however fascinating the math...

[@dlwatib]

+Lou Puls He at least remembered that it was a long name beginning with H. But is it so difficult to remember that it was an Irish mathematician in the 1800s and not a German physicist in the 1900s?

[@gfetco]

+Lou Puls Say my name!

[@morgengabe1]

Yourre mothers would all b so proud

[@ahmedgaafar5369]

i agree too.

[@calmsh0t]

Praise the age of digitalization. I can get all the knowledge I want from great sources and don't need to rely on local professors who can't explain even the simplest thing, plus I can filter out the stuff that university would want me to know but I never need for what I want to do. What a time to be alive!!

[@DrMerle-gw4wj]

Quaternions were created by William Hamilton, not Heisenberg. No doubt someone has already added this in the comments.

[@englishforfunandcompetitio248]

Aside from mistakes in mentioning History, the intuitive approach he has applied for teaching the subject, is better than many others on the KZbin.

[@APaleDot]

26:40 He says the quaternion ( cosθ, sinθ v ) represents a rotation by angle θ, but it actually represents a rotation by angle 2θ. The reason: when doing a rotation, you do a "sandwich" product to prevent the vector from being pushed into 4D space, u' = q u q^-1 which applies the quaternion twice, resulting in a rotation by 2θ.

[@zdspider6778]

Yeah, that's what I thought! It should be: _(cos(θ/2), sin(θ/2) * v)_ And he didn't explain the "sandwich" part... At least I don't think he did. And there's no "part 2".

[@JohnCena963852]

May not be perfect for some details, but definitely the best clarify of quaternion. Thank you sir. btw, does anyone know which OCW does this lecture belong to?

[@dendrogenhs]

This lecture skips details, and the presenter does mistakes, but he really gets the intuition: this is the easiest to understand video about quaternions I ve found so far...

[@JA-yi8bs]

A concept I was not taught at University and now faced with in my research. Your explanation has been so helpful for my understanding - thank you!

[@yunhyeokchoi2004]

8:36 humanity restored

[@realdeal968]

I watched countless videos on quaternions and this one is the best by far.

[@michaell685]

Per Wikipedia, not Heisenberg (1937-1976) but Rodriguez & Hamilton in the 1840s developed Quaternions. Hamilton was its great advocate. " Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840,[1] but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations."

[@slickwillie3376]

They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.

[@8cccpeevostokzempf]

Not too sure about the Heisenberg reference.

[@onetwoBias]

Excellent lesson! :) Impressive that he managed to make this comprehensible to someone with only a basic understanding of vector math in three dimensions, who has never heard of quaternions. (me)

[@timelsen2236]

Fumbled it, Hamilton not Heisenberg. Must be busy elsewhere.

[@benmansourmahdi9097]

professor i owe you for ever

[@MatheusLB2009]

The F1 Driver, not the Meth cooker

[@shohamsen8986]

Did he say heisenberg??? Waaaaaat

[@thetntm2]

+Shoham Sen it wasn't heisenberg. The man who discovered quaternions was sir William Rowan Hamilton.

[@shohamsen8986]

thetntm yeah i know hence the question mark... :)

[@comprehensiveboy]

That was terrible misinformation. I can only guess it is because some people are not too interested is who did what.

[@ChristianS1978]

+comprehensiveboy According to Simon Altmann (cf. Wikipedia) it was Carl Friedrich Gauss in 1819 (only published in 1900).

[@Beon234]

+Shoham Sen "I am the one who rotates" - Heisenberg

[@pavelperina7629]

34:00 please always remember original matrix, construct quaternion from original mouse position to the current one, construct quaternion (i guess there should be phi/2, but i'm not sure) and the convert it to model matrix. On mouse release store that model matrix. Otherwise they will be ugly artifacts caused by sampling of mouse coodinates and I guess rounding errors as well. PS: i have to find how to convert quaternion into 4x4 matrix, because it would be nice to visualize that in some projections. I always found q^bar * v * q as 3x3 matrix

[@NoisySoundFilms]

is there a second part of this lecture? i would like to a real application of how to move objects on 3D space. By the way! it has been a very great time seeing this lecture!

[@johntessin6398]

William Rowan Hamilton invented ( discovered ) them. There is a wonderful neighborhood in the area called South Park in San Diego called Hamiltons that specializes in micro brews. I find a twisted satisfaction in that for some reason.

[@MarincasChannel]

Great lecture! But I'm still confused why quaternions actually use θ/2 instead of θ to represent an axis-angle rotation. My brain reaches a gimbal lock when thinking about this.

[@BlueinRhapsody]

It is because to perform a rotation with quaternions on some 3-vector v, we take our unit quaternion p to get v' = p v p^-1. When we multiply p times v, we rotate on the unit sphere, but we also rotate into the fourth dimension [p v = (*-p dot v*, p_0 v + p x v)]. When we multiply after this by p^-1, we rotate back out of the fourth dimension by the same amount, and we also rotate forward by the same amount on the unit sphere. Basically, the first multiplication rotates us halfway there (and a little the wrong way), and the second multiplication rotates us the rest of the way there (and cancels out that 4D bit).

[@francescorizzi2601]

@@BlueinRhapsody please, if you can give any reference link to explain exactly this phenomenon it would be great. I'm struggling to understand this. Thank you!

[@BlueinRhapsody]

@@francescorizzi2601 Honestly, I just learned about quaternion rotation from Wikipedia: en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

[@phartatmisassa5035]

en.wikipedia.org/wiki/Quaternion#Matrix_representations Hmmm, So I was sittin on the porch tonight thinkin, and the following is the question I came up with. Given vectors U, V elements of R3 and a quaternion (say) Q element of H s/t Q is the quaternion which rotates U to V ( as with the track-ball), Is it possible to find Q' (Q prime), i.e. the dQ/dV, or the derivative of Q with respect to the change of V s/t V rotates to U. Would that even be useful?

[@richardfantz5694]

1. Maxwell's original 20 quaternions instead of the dumbed-down, truncated equations he and Heaviside later developed which is what everyone's taught in school + Nikola Tesla + Non-Herzian waves = Enough said.

[@gc1979o]

Heisenberg hahaha

[@abhinavkumarkumar3370]

Why there is -v1.v2 when multiplying q1and q2. @17 mins

[@mattwolf2887]

Really great lecture. Thanks :D

[@johnhefele5432]

Does anyone have these notes that the lecturer keeps referring to? If so, could you kindly share them?

[@lawrencedoliveiro9104]

Another useful feature of quaternions is that they interpolate very nicely, which is useful for animations. Say you have two orientations of an armature bone in your character. Each orientation can be represented by a quaternion. If these are keyframes, then the animation software can interpolate the intermediate orientations by interpolating the quaternions. This automatically gives you a uniform movement along the great circle connecting the two orientation points. If you were trying to interpolate Euler angles, then you would not (in general) get movement along a great circle. I think the actual curve might be a loxodrome (I’m not sure). In any case, it won’t look nice.

[@lunchen7985]

28:00 is the punch line if you're here wondering how quaternions can be used for rotations and for solving gimbal lock

[@Ybalrid]

I actually write good amonts of code using quaternions (because, 3D games and VR stuff) I never really fully understood what was these "4 numbers things", and how it can represent, well, rotation around an arbitrary axis, and why you multiply them togeter to get sucessive rotations, and all that jazz ^^"

[@stevel9678]

Quaternions were invented by Alexander Hamilton. Heisenberg was the meth kingpin on Breaking Bad. Glad I could straighten that out.

[@OlivierGeorg]

Good basic but approximative and incomplete explaination, which pushed me to search for more information: 1) Rotation by \phi around \vec(v) is given by q = (cos(\phi/2), sin(\phi/2) \vec(v)) 2) A position vector can be represented by p = (0, \vec(x,y,z)) 3) Rotation of p by q is given by quaternion operation p' = q * p * q^(-1). That operation is said to be computationaly cheaper than using matrices.

[@harandianr]

A little algebra would have made things simpler and cleaner.q=a+bi+cj+dk, Hamiltonian numbers are an extension of complex numbers. As person with mathematical background I found the lecture a bit confusing. In math you easily and cleanly show that Hamiltonian numbers form a number system but there is no commutation.Showing a=a+v part is the interesting part of this lecture. But I wish he had done it in a cleaner way mathematically. Wikipedia covers this but still I wish someone would post a lecture on rotation using Hamiltonian numbers in a detailed and clear way. But the way Hamilton wanted to make a number system in dimension 3 but it did not work. Mathematically , there are number systems of dime nation 1,2 ,4 and 8 and that is all, nothing beyond 8. Dimension 8 numbers are called Cayley With Cayley numbers you do not have a(bc) equal to (ab)c numbers. It took decades before Hamiltonian found their way into application . Let us hope that Cayley find their way in years rather than decades.

[@zdspider6778]

Hamilton invented the quaternion, not... Heisenberg. There's even a plaque on a bridge in Dublin where it "hit" him to use "ijk". He wrote it down as: _i^2 = j^2 = k^2 = ijk = −1_

[@TheRCrispim]

Hamilton, not Heisenberg. •-•

[@chanm01]

...now kinda wishing I had studied computer graphics in university instead.

[@mikedavid5071]

This is a great intuitive introduction to Quaternions. Knowing who invented quaternions gets you nowhere in understanding quaternions. Knowing the name means nothing. Knowing how to use them and forging new fields where they have practical use is quite useful.

[@tinkeringtim7999]

Thinks gimbal lock is a property of rotations rather than of Eulerian coordinates. Thinks it was invented by Heisenberg or something. Doesn't know why they were invented, especially that it was not a curiosity but specifically invented to address rotations in 3D space. There was even a Quaternion society, it was a massive fight in the history of mathematical physics at the time. It is impossible to learn anymore than the superficial about Quaternions without knowing these facts. I'm sorry but you can only learn method from this guy, and be wary at that. As for conceptual understanding, he lacks sufficient context to have it. In other words, if you mute the video you will have less misleading explanations attached to the algebra and geometry on the board.

[@s1va3209]

No . HAMILTON. Quaternions were NOT found by Heisenberg. This isn't a course on quantum data science!

[@englishforfunandcompetitio248]

This professor is very poor. I would say Extremely poor. Poor in the science history. The poor guy doesn't even know when and where Quaternion were invented and by whom.

[@VanNguyen-kx6gx]

Teacher is no good, made many mistakes. Don’t know what the quarternions.

[@suave319]

The professor seems really nice but he makes a lot of mistakes.

[@JackLe1127]

who tf cares?

[@suave319]

Jack Le I do. And anyone who actually wants to learn does.

[@JackLe1127]

Most of his "mistakes" are things that you can easily look up and not really necessary conceptually. He himself even said he wasn't sure about those bits of information.

[@suave319]

Jack Le Well, if you don't care if a professor makes mistakes, then good for you. I like not having to look up every little thing the professor states in case it might be wrong. Also, thinking Heisenberg invented quaternions when you're a university professor is inexcusable. He was obviously a meth empire king.

[@JackLe1127]

Suave Atore That means he pays more attention to the concept itself. Who came up with an idea is just a google search away. You can't just google some concept and immediately understand it. That's why we need him to help us understand the concept. Common knowledge is easy to obtain.

[@baruchba7503]

Best explanation of quaternions I've heard. Thank you.

[@zdspider6778]

25:33 Shouldn't it be: _cos(theta/2), sin(theta/2) * v_ ? 🤔

[@SowmyanarayananP]

Great! Thank you so much!

[@jairo359]

Im a dumbass and I can tell that this lecture is a good one, just watch it a few times over.

[@shivanshiverma8025]

Thank you for explaining with such an elegancy, sir! I've been stuck on this topic for a long time now, and finally you made me understand it 😁😁

[@andyeverett1957]

Much about quaternions just fell into place with your lecture, thanks.

[@geoffreygoldman1115]

Nice lecture. I have a much better conceptual understanding of quaternions.

[@aylasedai2317]

Hamilton?

[@arnostzeman569]

Can me someone please say how the fuk it rotates with sinTheta, cosTheta v ?

[@notevolverastontopensando]

"Against the day" by Thomas Pynchon

[@shreksnail5892]

This gave me a nose bleed xD

[@wisdomokafor9631]

I don’t get the multiplication part.

[@cesarjom]

He's not explaining the quaternion property correctly. What Hamilton did was define the base quaternions as satisfying the following condition: i^2=j^2=k^2=ijk=-1. From these relationships, you could then derive the products of mixed base quaternions, such as ij=k or ji=-k

[@Cold73]

Not Heisenberg but Hamilton

[@abj1203]

Which website he keeps mentioning?

[@AndreaCalaon73]

Please feed students with Geometric Algebra and not with complexes, quaternions, ...

[@ericzeigler8669]

This guy needs to spend 10 minutes before class studying his notes. Everyone hates Prof. Aaaaaaah, ehhhhhh, ummmm. Leonard Susskind does the same crap except he asks the class, "Is this right?" Leonard is a bona fide genius, so I'll cut him some slack. I hope the department chair doesn't watch this video. No tenure for you, Prof. Ummm.

[@SwapanChakravarthy]

If one tries to define the norm of complex and others the values of i-sq and j-sq etc is equal to (- ) 1.

[@tcioaca]

Well, Heisenberg is probably the biggest inaccuracy. I would like a more solid explanation of what "gimbal lock" actually means. In this lecture, the gimbal lock is explained as if it were originating from another source of singularity inducing factor (alignment of two spatial vectors if I get his intuition correctly). A better approach to understanding gimbal lock is to _explain_ how the gimbal mechanism works. Students usually invoke gimbal lock every time their rotations work incorrectly or stumble upon a singularity.. which is not always caused by this phenomenon.

[@o_2731]

The Camera Man ← → ← → ← → ← → ← → ←

[@soda9197]

What course is this a part of?

[@mecdos]

this is really hard because i'm paying close attention to find out he made a mistake. and now i have to unlearn his mistake plus learn the correction. this should have been edited before uploading and this guy should have practiced his lesson before hand.

[@the_nuwarrior]

¿it can be generalizated to a 2^n- dimentional object?, ¿ exist an n such that it forms a cunmutative field ?

[@McTofuwuerfel]

Even he said Heisenberg, I am certain it was Hamilton.

[@williamolenchenko5772]

Some people heard "Hamilton" and some heard "Heisenberg."

[@micka6288]

At 19:55 why is division NOT inverseDenominator*numerator in that order like matrix inverse

[@Supercatzs]

Quaternions start at 7:07

[@justbeyondthemath4559]

Quaternions are the first step to fixing Euclidean space. To the beginner, you can think of i,j,k as 90 degree rotations in the respective planes. Just like the Argand plane (complex plane) or i plane in the quaternions. 1xi = i ixi =-1 -1 x i = -i and -i x i = 1 which puts us back to where we started. BTW I right multiplied to show you next state but technically it should be left side.

[@pianochannel100]

Go play with 3 blue 1 brown's interactive video lectures if you want to learn about quaternions.

[@ogunfidodoadekunle2807]

I find quaternions applicable to statistics,also find useful the idea of (cosx+sinx.v) where v is a unit vector.

[@diabolicallink]

Everyone is complaining about him using the wrong name. But this isn't a history course, so does it really matter who?

[@meriquirogaalbarracin2420]

God bles you bro❤😊😊😊

[@lucyfrye1337]

Thanks, that was good. But I wasted time because of his mistakes. He should prepare his blackboard work more.

[@definesigint2823]

[breaks chalk] Well, somebody obviously supplied this classroom with right-handed chalk.

[@TheLazyKey]

Great video on quaternions. I still don't quite understand them fully. But I'm sure applying them practically will help me fill in the gaps.

[@yiyangtang3622]

This is an clear explanation about quarernions, thanks a lot

[@brendawilliams8062]

It’s turning too

[@iraqplayer7270]

In case someone is looking for the cheat sheet the professor is referring to: graphics.cs.ucdavis.edu/~joy/ecs178/Transformations/Quaternions.pdf

[@pierresarrailh6617]

thanks a ton I really needed it and cant access the site as I am not a student

[@iraqplayer7270]

@@pierresarrailh6617 You are welcome! So you have gotten the cheat sheet right?

[@iraqplayer7270]

@1 conscience 0 dimension Good to hear! and yea, I searched up the matrix, it seems interesting.

[@crafteurG]

He watched too much Breaking Bad, it's not Heinsenberg it's Hamilton !

[@MykelGloober]

So is the V value equal to the pitch, yaw, and roll? Or is that just the vector value? Can anyone point me to a lecture that talks about vector math?

[@bsergean]

Great presentation

[@vwcanter]

This is a valuable introduction, for people like me, who need to get started on these.

[@gerardoconnor4278]

William Rowan Hamilton Trinity College Dublin - discoverer of quaternions

[@rasitcakir9680]

Engineers! They get what they want. They don't care where they come from.

[@tedsheridan8725]

People who say OK after every sentence are infuriating.

[@brendawilliams8062]

There is some more leaving the 1001032155

[@vgrinberg1]

Hmm, wasn't Heisenberg a math cook? Lol

[@yb801]

4*4 matrix? Why ? Shouldn't it be 3*3 matrix?

[@miltonlai4850]

Easy to understand, very good explanation.

[@KunalShah62]

Where did the 5th term in quaternion multiplication come from?

[@cyborgbeingadroidthinklike5737]

His attitude of teaching shows that he is very much conscious about his topics

[@cyborgbeingadroidthinklike5737]

His attitude of teaching shows that he is very much conscious about his topics

[@liamcjbeistle3274]

William Rowan Hamilton used for navigation gimbals, simulation motion platforms etc

[@zeeshanijaz2870]

Around 10:00 the professor says that Heisenberg was not able to figure out ij and was forced to add another term dk to tackle the problem.Well my question is if the assumption we make is that i square = -1 and j square = -1 then it follows that ij = -1. So it is not undefined. So there was never even a problem to start with. Can somebody answer this please

[@abeno62]

+Zeeshan Ijaz I am no mathematician, but I don't see how you can infere that ij equals minus 1. With the assumption that i^2=j^2=-1, we only can say that i^2 = j^2 nothing more. If I follow your path, you would end up with i=j and then it's completely useless because you only get 'simple' complex numbers.

[@HeliosFire9ll]

+Zeeshan Ijaz I've come with the same conclusion, did you ever find the answer to this question?

[@maxwibert]

1^2=1 and (-1)^2=1, yet 1*(-1)=-1. so i have a counterexample to the argument "a^2=c and b^2=c implies a*b=c."

[@HeliosFire9ll]

ok this makes sense now thank you.

[@seven9766]

The Sentence is : i^2=j^2=k^2=ijk=-1