Fantastic Quaternions - Numberphile

[@rudymartin8583]

If you're like me and were completely confused by the equation i^2 = j^2 = k^2 = ijk = -1 , go to the extra footage, it really helped me. The reason it looks so weird is because you lose the commutative property when you go from 2D rotation to 3D rotation, the property stating that ab = ba . This means that the order of multiplication matters, and that if you reorder them, you get a different result. If you'll imagine for a minute, when you rotate an object in 2D space, you can do more than one rotation, and the order of those rotations wouldn't matter; it'll end up in the same ending position. But if you're rotating an object in 3D space, then the order of the rotations absolutely matters! Turning an object 90 deg counterclockwise then 90 deg away from you (if that makes any sense...) is not the same as turning it 90 deg away from you then 90 deg counterclockwise.

[@jackismname]

Marie Alexander Thank you very much for this explanation!

[@ceruchi2084]

Incredibly useful comment! I just spent the last five minutes rotating a beer can in "two" dimensions and then in three dimensions, and now this makes intuitive sense.

[@ΘωμάςΣέλμπυς]

ceruchi same with my finger😂

[@mawebb88]

I did the same with a nail file and I had the revelation like you.

[@xavierstanton8146]

Yes, you could say that the quaternions aren't Abelian.

[@pegy6384]

I would pay to watch a blooper reel or outtakes from all the main Numberphile presenters. I'm sure there's a lot of funny stuff that we never get to see.

[@numberphile]

+Peg Y I think they'd pay me NOT to show that! :)

[@davecrupel2817]

+Numberphile I'd pay for a dvd of them 😁lol

[@VicvicW]

+Numberphile We shall outbid them!

[@moritzkockritz5710]

+Numberphile We are MORE!

[@hugo.teixeira]

+Peg Y I imagine them committing mistakes in easy sums or basic stuff like 4+2

[@PhilippeGouin]

Finally an video that doesn't start with "it's complicated, so just ignore what they are, here's how to use them"! Great explanation, thanks!

[@AlanKey86]

I had a friend at uni whose answer phone message was: *Sorry, the number you have dialed is imaginary. Please rotate your phone through 90 degrees and try again* Epic.

[@icannotchoose]

I aspire to be that person.

[@VicvicW]

Byootiful!

[@NickiRusin]

+AlanKey86 That dude was probably a real fun dude to talk to.

[@U014B]

+Drama_Llama_5000 I am that person!

[@NickiRusin]

***** You're cool, man.

[@gabrielkwiecinskiantunes8950]

6:01 James realizes he can't express his thoughts using mere words.

[@DeathBringer769]

I'm sure many mathematicians go through the same feeling many times in their lives, lol.

[@TheNefastor]

And that's why we have numbers :-)

[@HeavyboxesDIYMaster]

6:03 What math technique is that?

[@the-keymaker]

It's a mathematical gesture, duh

[@YourMJK]

It's the gesture equivalent for crossing out your previous result

[@the-keymaker]

+YourMJKTube genius

[@karanbajwa1963]

+Heavyboxes Its a 'mis-calculated' gesture.

[@TNBLUEDIXIE]

Noice

[@skroot7975]

James is like a kid at christmas all the time. Love it!

[@Omedalus]

I've been watching dozens of videos on quaternions. This is the first one that actually explained how quaternion multiplication results in spatial rotation. Specifically at time 4:50, In showing how to rotate by 45° on the complex plane, everything else just kind of falls into place. Magical. Thank you!

[@krakow10]

My origin story! I cry every time.

[@TheNefastor]

You're got a... complex story ? 😅

[@lawrencedoliveiro9104]

Peeling quaternions can apparently do that to you.

[@onigumo]

KZ, man of culture

[@vin_fm2354]

This entire comment section makes me wanna d(i)e

[@asheep7797]

He still keeps his name after 6 years!

[@brianhoskins1979]

Numberphile is easily the best channel on all of KZbin. When I was in school I feared maths, and I gave myself the idea that I was really bad at it. I have since learned that actually I'm *not* bad at it - I just needed to have a bit of confidence and the will to try hard, that's all. If Numberphile had been around when I was in school I think it would have been the inspiration I'd have needed, and maybe I never would have given myself the stupid idea that I couldn't do maths.

[@Entropy3ko]

"lots of i" love that Brit-speak. You explained it very well James!

[@JacobFelten]

Imagine if the video ended right after he said, "Their fantastic!" at 0:35. He's so charismatic that I still would have thought it was a great video.

[@komrad36]

I use quats every day! In addition to computer graphics, they are useful in aerospace. I use them for satellite attitude control. Cheers!

[@-Danny]

You have an awesome sounding job! I bet it pays well. Haha.

[@thegrumpysock6391]

what is your job and whar did you study?

[@MattCattrell]

Maybe I'm just tired and didn't catch it in the video, can you explain what each of the four numbers represents in the real world? I just thought it through, using Kerbal Space Program as my mental model, is this correct: One number for "forward and backward" one for "side to side", one for "up and down" and one for "pitch and yaw"? I think that makes sense now... Maybe I need a nap and then I should watch this again... Thanks!

[@EebstertheGreat]

+komrad36 I'm curious, when is it easier to use quaternion multiplication over rotation matrices?

[@komrad36]

+Yassine Sayadi I'm still in school, actually. Dual degrees in aerospace engineering and theoretical physics. But I also work at NASA doing basic orbital and attitude propagation simulation and research and also work as an independent contractor for the NSF, doing satellite attitude determination and control (ADACS) for a satellite constellation called TRYAD. I write the software that my colleagues use to model satellite behavior in orbit.

[@2Cerealbox]

I had been hearing that quaternions were scary, even though it was how all game engines work underneath the surface. You've made the subject much more approachable.

[@fakjbf3129]

Charles Lutwidge Dodgson was a mathematician at Oxford when this was discovered. He hated abstract math, and thought it was all a bunch of hogwash with no basis in reality. So he wrote a book in which he included a caricature of various abstract mathematical concepts, including quaternions. He wrote it under the pseudonym Lewis Carrol and it's called Alice in Wonderland.

[@voidisyinyangvoidisyinyang885]

"real numbers are a convenient fiction" - Bertrand Russell

[@bobnavonvictorsteyn9017]

That is the coolest thing I have heard all day

[@rosiefay7283]

What's the correspondence between various abstract mathematical concepts, including quaternions, and anything in Alice's Adventures in Wonderland? (BTW take another look at the name of the author in that book.)

[@grandpaobvious]

doubtful anecdote is doubtful

[@ragnkja]

@@grandpaobvious It’s the tea party scene, and in particular the bit around “At least I mean what I say.”

[@michaelgodfrey4911]

I have to admit, I've got a bit of a crush on James. He's got such a charming voice and friendly personality, he comes across as a really nice guy you could chat with for ages . Also, I just love the way he acts when he gets excited about maths as well, the way he lights up and can't wait to tell us the next bit and he's almost bouncing with joy. Not a bad looker either on top of that

[@youtubesuresuckscock]

That's what happens when Radiohead's signer teaches you maths.

[@iris5403]

@@youtubesuresuckscock Thomes Grorke

[@lohphat]

How can a 12min video finally get the concept of "i" across so clearly which my high-school teacher could not? I suspect I wasn't paying attention or I was just not interested or motivated to learn. Now it makes much more sense. Perhaps the development of mobile tech and graphics showing these concepts in real time make it easier...

[@swaggercr7]

What's so hard about understanding it?

[@MattCattrell]

+Longarmx This is something that just wasn't really dealt with well when I was in school, either. The concept of visual learners was well established, and no one would dream of trying to teach a toddler something without visual aids, but math teachers in the 90's, heck most teachers in high school, just seemed to forget this all and keep trying to explain using the same words and phrases over and over again... I hope that has changed these days!

[@lohphat]

+SwaggerCR7 As he stated, the term "complex" is misleading and the mechanical formulations are all I remember from school. The graphical relationships demonstrated here made it all "click" and come together.

[@RylanEdlin]

+lohphat I think the confusion is that modern usage conflates complex and complicated when they actually have different meanings: complicated meaning difficult and complex meaning built of many parts.

[@Theraot]

+lohphat As far as I can tell, most teachers will start by going about the square root of negative one. While starting at using them as vectors is easier to grasp.

[@Omedalus]

Thanks!

[@GamesFromSpace]

I've been using quaternions for years, as a programmer, and have absolutely NFI how they really work (even though I can implement them, and certainly know how to get results from them). Matrices, please.

[@hanniffydinn6019]

You have to get into algebraic geometry to truly understand then. Which means every number is really just a higher dimensional object in space. Real and complex numbers are just "cut" down views to what is really going on. Hence, we live in a multi dimensional universe !

[@eideticex]

+Joshua Pearce It's not too hard to understand if you plot out your axes that form from transforming the world's forward, up and right vectors against a quaternion. It's really not any different from axis-angle, some of the math is already done for you with a quaternion. Even with a matrix you wind up needing them to prevent gimble lock if your expecting unpredictable rotations performed on the matrix (like a camera being moved around).

[@GamesFromSpace]

(And every time I mention quaternions being something I don't grok, I get a new set of attempted explanations to add to the pile...) Alan Hunter You mostly just described how and when to use them, which is something I've been doing fine for years. And I would never use quaternions AND matrices, I'd use just matrices if I needed that extra info, and convert to quaternions only when a different API needs them, or it somehow simplifies a function. I guess it depends how you define the term "use" in this context. They certainly both have their uses.

[@frankschneider6156]

+Hanniffy Dinn Only if one assumes these numbers have physical reality. Mathematics is a tool to describe physics/reality. Reality is not the implementation of what's mathematically possible.

[@yuriythebest]

+Joshua Pearce Yess sir!! So I'm not the only one! Euler angles are fine, but quaternions are all about trial n effort in getting your desired result. Watching this video, I felt as if he was using elvish script to describe something I use - it's not us who are wrong, it's the maths people with their complex stuff!!

[@firefoxmetzger9063]

This video literally explained in 12 minutes what my professors failed to explain for 6 month. Thanks so much for this!

[@ConstantlyDamaged]

A video explaining why the extra dimension is needed, would be awesome. Also, if my calc3 professor at university had spent the 12 minutes to explain what complex numbers were, the way you did, I might have actually completed the damn course >.>

[@Vulcapyro]

+Darthane ...You took a Calc 3 course without some linear algebra?

[@ConstantlyDamaged]

Vulcapyro I took it in order, never got any advice to do otherwise. If there was something I missed then that explains my freakout over completely not understanding what was going on :)

[@logicalfundy]

+Darthane I believe it has to do with 2D having one degree of freedom (you just rotate around a point), but 3D having three degrees of freedom (you can rotate around the x, y, z axes of a point in 3D space).

[@ophello]

+Darthane You need three pieces of information to define a point in space, but that's not enough to tell it what to do. Quaternions are about manipulating points in space, not just defining their position.

[@DFPercush]

+Darthane Think of the "stab and rotate" analogy. What do you need in order to "stab" something? A vector, right? Imagine the pointy tip like a spear piercing the object. A vector in 3d space takes 3 numbers. Then you need another number for the angle.

[@Hewlett-Packard-Lovecraft]

What a brilliantly rendered explanation. Whenever I learn of mathematical functions this beautifully visceral, and ethereal, it's not unlike being violently shaken awake from a deep slumber and it submerges me back into the unsettling divinity of sacred Mathematica. As above, so bellow

[@r6inhardt]

Excellent, thanks for sharing this. James did a great job of helping me to follow the math for Quaternions.

[@pidigi]

You have no idea how this channel is helping me to grow. I’ve no enough thank you

[@TecaLucas2ndChannel1]

Quaternions sound like a race out of a Sci-Fi movie.

[@johnhadaruga1375]

Nemes in Sci-Fi movies came out of science . Duh :D

[@Spedley_2142]

Way more complex than Bynars!

[@mikekuppen6256]

Man I love Dr Grime! Even if I don´t always understand him right away, it makes me happy to look at him being enthusiastic.

[@Madsy9]

Quaternions is the way to rotate 3D points in pure mathematics, but in practical engineering and software development, Euler angles are also popular. The advantage of euler angles is that it's easy to understand as a set of combined 3D rotations. The advantage of quaternions is that it's only one single operation, so you avoid gimbal lock; a problem which haunts euler angles unless you design around it. But you can also combine both techniques. For example, in computer software like Blender and Maya, a user could specify rotations by using euler angles, which later is converted into quaternion form to avoid gimbal lock. And even later combined and turned into a 4x4 affine transformation matrix. Also notice that the quaternion rotation as described by James, "hph*" is a very formal description of the rotation. Which is interesting (math is interesting!), but not very practical. In practice you would combine multiple quaternions together, turn the final quaternion into matrix, and then do a matrix-vector multiplication. Regarding the "control" you lose as you go up in dimensions: Octonion and sedenion multiplication is neither commutative nor associative. I would love if Numberphile could make a video on this question though: Why are the most useful algebras of a dimension 2^N? It is a natural consequence of applying the Cayley-Dickson construction, but *why* do algebras of these exact dimensions (1,2,4,8 ..) have nicer properties (or is defined at all!) than say R^3, R^5 and R^7? Is conjunction undefined in the latter dimensions?

[@warlour]

Thank you so much for still having this old video! It really broadened my understanding of Quaternions and Complex Numbers :)

[@piticea]

YES, a video with james grime :D

[@radenzito]

Omg Dr James Grime. This guy has born to make others learn and understand. Thank you a lot for your work

[@franciscomorilla9559]

*New Numberphile video* Let's see it *James is in the video* YEAAAAHHHHHHH

[@spazmobot]

James - your enthusiasm about math makes what would probably be a boring subject to most people, really interesting. Thanks for making it fun!

[@TheScabbage]

Yaaaay finally, quaternions :D So I watch most KZbin videos at 2x speed... 6:01 Glorious.

[@Garvish_Parekh]

Finally a great video, after spending soo many days reading blogs and watching tons of videos which were so complicated from the get go, still bit confused about 3D rotation, but I am much more clear and have basic understanding to understand more complex material.

[@bazookah187]

I'm a simple man, I see Dr. James Grimeon in the thumbnail, i click

[@Don-h4d]

You are just a real idiot attempting to be funny by consciously playing the role of an average idiot

[@damienjones1487]

@@Don-h4d dang, what an absolute intellectual you are

[@Don-h4d]

harold the alien I appreciate your compliment and effort, considering you even had to edit a one line sentence.

[@damienjones1487]

Wow, that really hurt my feeling

[@damienjones1487]

@@Don-h4d im sueing

[@tmd4951]

I listen a lot of videos to understand what quaternion is. And let's me say this is the best video that explain very clearly.

[@MrMakae90]

Please, a video on Octonions. Thanks.

[@melficexd]

And who in the name of the higher realms need an 8 dimension vector?! 😂 *sits in the corner of a tesseract*

[@stulora3172]

@@melficexd physicists

[@scififanman]

Being a CAD technician I found this video very insightful. After I went to college to become a CAD tech for mechanical engineering/design, I discovered all the work I was doing involved 3D, whereas older, more experienced techs often only worked in 2D. All of my work involved modeling in 3D, and some of the CAD programs like Alibre were very odd about how they manipulated things in 3D space. But knowing this form of math helps explain some of the operational quirkiness certain CAD programs have, when you're working in 3D.

[@theatheistpaladin]

Quaternions... Octerions... Just sounds like alien star trek races.

[@egor.okhterov]

Octonions :) 'Octo' means 'eight'.

[@metleon]

+Охтеров Егор Still sounds like an alien race.

[@kaibeasley5965]

Are there hexadecions?

[@aaronhultstrand2919]

+Kai Beasley As ZalgoOfficial said they are called sedenions. But interestingly octonions are the highest dimension complex numbers that adhere to a division algebra. There is an infinite number of dimensional algebras going passed the octonions, such as he sedenions, but they are not division algebras...i.e. they have no zero divisors and are different beasts than the normal algebra we deal with.

[@picknikbasket]

This one of Jame's best vids, really interesting and the extra footage leaves me wanting even more!

[@swaggercr7]

So how many components for rotating in a 4 dimension? ...8??

[@yxlxfxf]

How about Graham dimensions?

[@johngalmann9579]

+SwaggerCR7 I would have thought it was on factor for scaling and one for each plane of rotation, but no I wonder what the system is...

[@yxlxfxf]

John Galmann seems like 2^(n-1) for n dimensions,idk why though...

[@JackProudfoot]

+Famfly So a zero'th dimension requires half a component? Interesting

[@twilightknight123]

+SwaggerCR7 As said in the video, the next level is the octonions (8 components) which are quite useful in physics for describing the motion of a spin-1/2 particle (specifically the split octonions, though the octonions do have their uses with symmetry). +Famfly I believe for n rotating spacial dimensions, the pattern is an n-2 dimensional object to rotate around (2D plane around 0D point and 3D space around a 1D line) and 2^(n-1) components as that's the size of the smallest extension of the arithmetic ring.

[@chrisb3380]

Great to see you back James. Hoping for more vids from you in the future!!

[@cubedude76]

quaternonions are great in fajitas

[@brianbethea3069]

I prefer dicedonions thanks.

[@RyanGatts]

This is probably the first time numberphile has answered a question I have actively been wondering, and I love that the answer is so cool!

[@omrialkabetz5602]

I don't understand something about the equation i^2=j^2=k^2=ijk=-1 : If you square ijk, will the result be: (ijk)^2 = (-1)^2 = 1? Or (ijk)^2 = (i^2)*(j^2)*(k^2) = (-1)*(-1)*(-1) = -1?

[@jeffreyhersh908]

+Omri Alkabetz Not quite. What it is saying with ijk = -1 is ij = k. Since i^2 = j^2 = k^2 = -1 this mean that i,j, and k anticommute, that is ji = -k. So (ijk)^2 = (ijk)(ijk) = (k^2)(k^2) = 1. Or (ijk)^2 = (ijk)(ijk) = (i^2)(jk)(jk) = -(i^2)(j^2)(k^2) = 1

[@snowprinceintardis]

+Omri Alkabetz They are not actually just numbers, but they are described as 'root of -1'. They are really just unit vectors that tell us to go which way and how much. Since it is a 3 dimensional vector multiplication and they have 90 degrees between them, i and j multiplied 3 dimensionally would be ij=k. and the others i=jk, j=ik. What happens with taking the square of that is actually you've changed the parenthesis, and therefore changed the priority. It's not just simple mathematics anyone learns in highschool, it's rather more complex. But I am in my first year in college, so I may not be completely correct in explaining. I did not understand the i^2=ijk=-1 at first either.

[@1ucasvb]

+Omri Alkabetz (i j k)^2 is not (i^2)(j^2)(k^2). (i j k)^2 is i j k i j k . You can't move the i's together to make i^2, for instance, because this property of "reordering" (commutativity) had to be given up in order to construct quaternions as a number system that is consistent.

[@douggwyn9656]

+Diego The Star Pirate (ijk)^2 = ijkijk = (k)ki(i) = (kk)(ii) = (-1)(-1) = 1. For octonions it's harder to do the algebra, because not only are they noncommutative but also nonassociative.

[@Pianothegamer]

+Omri Alkabetz I don't understand this properly either but I can tell you that: (ijk)^2 doesn't equal to (i^2)(j^2)(k^2) because (i^2), (j^2) and (k^2) are all equal to ijk So, (i^2)(j^2)(k^2) is actually equal to (ijk)^3. I believe that your first equation: (ijk)^2 = (-1)^2 = 1 was correct.

[@robertwatson4840]

Came here because I am reading Against the Day and this was a perfect explanation for a mathematical neophyte like myself who never got further in school than pre calculus. Thank you so much!

[@tomasouzaheuert]

A + Bismuth + Carl Johnson + Donkey Kong

[@MeteOguc]

This was such a clear and useful lecture! I should have learned about this relation about 10 years ago! Thank you!

[@lawrencecalablaster568]

Anyone else think that James is awesome? :)

[@mnada72]

I had no idea that it's possible for me to understand this topic that easily, and this proves that everything is understandable as long as there is the one who can explain it

[@CatnamedMittens]

He's alive!

[@ShayBowskill]

Has anyone else noticed how much the very slight camera shake contributes to how cool these videos are? Makes such a huge difference if you imagine how this would be without it

[@DragonMasterClay]

Why couldn't you have posted this one year ago. Last spring I was in Dublin, I would have gone to see the bridge!

[@sampadabhatnagar5818]

I really liked the way you did storytelling to explain the entire concept, makes it more fun to learn!

[@culwin]

Both i and j = sqrt(-1)! Finally mathematicians and engineers can find some agreement.

[@masicbemester]

factorial?

[@xinpingdonohoe3978]

@@masicbemester no

[@Titanknox]

thank you so much for this video, ive been doing lots of video game programming recently, and i thought i new quaternions, but this demonstration completely locked it down.

[@petros_adamopoulos]

Been under that bridge on pilgrimage before it was cool.

[@picklesnorf101]

This video was awesome! Thank you so much for posting this. Numberphile videos are always the highlight of my day.

[@tub944]

Do a video on TREE(3) please

[@branthebrave]

+tub944 They did.

[@tub944]

+Brandon Boyer which one?

[@U014B]

Fiddy?

[@ophello]

+Brandon Boyer No they didn't. They mentioned it by name once but they never analyzed it.

[@anticorncob6]

+Diego The Star Pirate It isn't. I don't think there would be a problem with a numberphile video about it.

As a 3d artist who rotates stuff in 3 dimensions everyday, I can say I felt in love with you and your passionate way of explaining quaternions

[@robertm.6243]

i really want him as my math professor...

[@EJM07]

This is by far the best video on quaternions ever.

[@sethgartner5057]

Unless I completely misunderstood, the "i" he refers to represents the square root of 1, but what do "j" and "k" equal? Like from a math standpoint, or are they just variables following "i" in the alphabet?

[@AnthonyMata]

i means imaginary number and its square root of one.

[@ThunderWorkStudioAMGE]

+Anthony Mata square root of minus one

[@ben-devries]

+Anthony Mata i is the square root of -1, the square root of +1 would be a real number

[@AnthonyMata]

+ThunderWork Studio yeah I forgot the neg sign since sqrt of one is one lol

[@sethgartner5057]

+Ben DeVries yeah I realized i messed up, i is the square of -1, but that still doesnt answer what j and k are

[@owaischunawala4030]

What are the properties that we lose (11:50) as we go a step further i.e. increase the dimensions?

[@UCH6H9FiXnPsuMhyIKDOlsZA]

...Why can't you just use 3 -- yaw/pitch/roll? Or does that only describe rotational position (that's not the word is it) as opposed to a rotation?

[@komrad36]

+TheNewbiedoodle You can - those are called Euler angles and are indeed the smallest (least memory required) way to represent an attitude. They have some disadvantages, however. I wrote a similar answer in depth as a reply to EebstertheGreat above. Cheers!

[@JasonEwton]

+komrad36 With Euler angles, I think you run into something called "Gimbal Lock"?

[@Thomcat]

you also need to define your rotation order when expressing a rotation

[@GMLscripts]

+TheNewbiedoodle To add to what the others have said, quaternion interpolation also rotates an object with a natural-looking, uniform angular velocity, something that could be very difficult to achieve with yaw, pitch, roll.

[@olafv.2741]

Great video. A nice demonstration of mind over matter: you stopped the clock at 16:00.

[@davejacob5208]

is there actually a proof that shows that it is impossible to make the rotation with threedimensional numbers?

[@uelssom]

Dave Jacob you need 3 rotational axis to fully rotate a 3d object. The "imaginary" (quotes bc i dont know how they call j and k) provide such axis.So you need 3 imaginary, plus the real = 4 dimension numbers

[@davejacob5208]

uelssom is there a proof?

[@6exG]

google 'gimbal lock''

[@hichamelyassami1718]

in robotics it's called 'wrist flip' or 'wrist singularity', it is when the path through which the robot is traveling causes the first and third axes of the robot's wrist to line up, the second wrist axis then attempts to spin 180° in zero time to maintain the orientation of the end effector...the result of a singularity can be quite dramatic and can have adverse effects on the robot arm, the end effector, and the process.

[@hichamelyassami1718]

In formal language, gimbal lock occurs because the map from Euler angles to rotations (topologically, from the 3-torus T3 to the real projective space RP3) is not a covering map - it is not a local homeomorphism at every point, and thus at some points the rank (degrees of freedom) must drop below 3, at which point gimbal lock occurs.

[@simetry6477]

Sometimes I feel like am jousting with you, but thankfully you bring it back to a point.

[@ragnkja]

So to include another dimension of rotation, you need twice as many terms.

[@douggwyn9656]

+Nillie It depends on whether you just want the new location, or the orientation and the location.

[@suremarc]

Not exactly. The first thing to note is that rotations in the complex plane are done via multiplication, whereas with quaternions one takes the (group theoretic) conjugation by an element 'q', i.e. x rotated by q is qxq^-1. With octonions you get 7 dimensional rotations, although there are some caveats.

[@kevincarmody1207]

+Nillie no, in 4D you only need quaternions, but with fewer restrictions that in 3D. See my reply to +SwaggerCR7.

[@kevincarmody1207]

+Matthew Cramerus group theory is not necessary. Quaternion multiplication is an extension of complex multiplication in exactly the same way that complex multiplication is an extension of real multiplication. The notation q^-1 means reciprocal in quaternions just as it does in complex numbers, whereas the conjugate is different and is denoted q*. The conjugate is used for 3D rotation, as the video explains.

[@suremarc]

3D rotations via quaternions are given by inner automorphisms of H, which form a group under composition.

[@9000fail]

This channel never fails to blow my mind

[@whitherwhence]

For your complaint abput the name "complex number", in Hebrew we basically call them compound numbers. So, ha. Neener neener.

[@krakenmetzger]

Yeah but in England we have foreskin. Neener Neener

[@denizaydinmusic]

You are an angel man!!! The clearest explanation i've ever seen.

[@mighty8357]

Anyone know the exact reason why a 3-dim number is not enough? Dr Grime just said that we need a 4th dimension without say why (other than it wouldn't work if we have 3)

[@komrad36]

+Phil Diesch You can use just 3 - those are called Euler angles and are indeed the smallest (least memory required) way to represent an attitude. They have some disadvantages, however. I wrote a similar answer in depth as a reply to EebstertheGreat above. Cheers!

[@eideticex]

+Phil Diesch It comes down to defining space. In 2 dimensions there is only a positive and negative rotation you can perform, clockwise or counter-clockwise. The axis is already defined by the topology of the space. In 3 dimensions however you can rotate around all axes freely, so to perform a rotation you must defined which axis to rotate around and how far to rotate. Notice in the math if you leave out sin(theta) you have coordinates, draw a line from 0 to those coordinates and it forms the axis that the quaternion will rotate around.

[@DreadKyller]

+Phil Diesch 3 dimensional numbers can represent a rotation in 3d space, however there's a few oddities with it: 1) The order of the rotations matter, if you rotate along the x then the y then the z, you'll have a different ending rotation than if you rotate along the z then the x then the y. 2) You can't easily rotate around a specific line, with quarternions you can place a line through the object and rotate the object around that pole, no matter the angle of the pole, with eulers you're limited to rotating on the 3 dimensional axis, and thus doing the same is complicated and ends up being the same math as is behind quarternions.

[@Samudbhava]

does that mean that in 4D there are 8 axes?

[@FeaturingMaxAsMax]

The exact reason is that rotations in 3-space form a 3-dimensional Lie group, called SO(3). The elements of this group are 3 x 3 matrices, hence they appear to depend on 9 parameters. However, it turns out that the matrix representation is really wasteful. The slicker way is to realize that SO(3) can be double-covered by the unit sphere S^3, which sits in 4-space, so you *only* need 4 numbers. I think that Dr. Grimes should have said it that way. The miracle is not that you need 4 parameters instead of 3 -- the miracle is that you need 4 parameters instead of 9. A slightly more handwavey way to say it is that a rotation is defined by four parameters: an axis of rotation, given by v1 * i + v2 * j + v3 * k (which has three parameters) and an angle of rotation, given by theta (the fourth parameter).

[@Unidentifying]

Mr Grimes is an absolute legend, love his charisma

[@iTzDeyo]

Who else is came here after watching Joe Rogan and Brett Weinstein talking about this??

[@charliesmash]

iTz DeYo right here

[@leoarnstein9444]

Eric Weinstein

[@ChuckDownfield2727]

Eric not Brett mon frere

[@b.f.skinner4383]

except here Dr Grime doesn't attempt to over complicate this using esoteric language and far fetched analogies

[@johnmeo1532]

Explanation was simple and very clear all the way up to the point he introduced the quaternions. I was hoping to have a full understanding of these elusive beasts but I still couldn't grasp all of their properties. He devolved from an intuitive explanation into a very formal representation that is only understood by people that already know what quaternions are about.

[@Durakken]

I can follow you until you get to the quanternians...cuz you explain what they are but not how it works... It's kinda annoying cuz I'd really like to understand how the math works to go along with the understanding of other parts. Something tells me that if I had learned trigonometry in HS I'd know this or if I remembered what cosigns and tangents were this would be a lot easier to get. The sad this is, to me the whole need 4 number thing to me is obvious once the first part is explained but you seem really excited about it that this was such a hard problem.

[@egor.okhterov]

I don't know if it makes anything easier, but here we go. Imagine we have a point on a plane and we want to know how it will be seen on the coordinate axis, if we know its angle of rotation. From the perspective of the axis X this point has moved cosine distance from the origin. From the perspective of the axis Y this point has moved sine distance from the origin. Now, if we magnify cosine on the X axis so that it becomes equal to one, what will happen to sine? The answer is the sine becomes equal to the tangent. If we scale sine instead, the cosine will become cotangent.

[@MattCattrell]

+Охтеров Егор uh, I'm not the guy you replied to, but since my highest math was algebra 2 around 15 years ago, this didn't really help... Not sure you can explain all that succinctly without pictures... Thanks for trying though!

[@Durakken]

That suffers the problem that most people have. You try to explain it in a way that someone who already understands it understands you, but anyone else won't. Anyways... Am I to believe that i, j, and k mean 90 degree rotation, or more specifically move along the "dimension" that is represented by that given terms... The first being x, second being y, third being z (depth), and fourth being rotation. I don't get the usage of i otherwise or the "multiplication" he uses either, but I only loosely listened there and know that multiplication can do things that look odd on the surface.

[@Durakken]

Matt Cattrell Similar. I only took algebra cuz I was forced to. I didn't find it hard, but I hated it and did as little as possible...and several times less than ^.^ The biggest issue I find with math in general is the terminology and the lack of connection to an example that makes it easy to understand and in this case it is unfortunate because it has an obvious practical example here.

[@zh84]

This video isn't really enough to explain how it all works. If you want more detail, I recommend an excellent book called "Yearning for the Impossible", which has a chapter on how quaternions were invented and, in particular, why they have to have four dimensions. Briefly, if you have only 1, i and j, what is i * j? Quaternions get round this by having i * j = k, and so on.

[@NoriMori1992]

6:01 - Maybe I haven't watched enough James videos, but that's the silliest thing I've ever seen him do and I love it. XD

[@sohee7597]

6:02 is he speaking in 4 dimentions?

[@LeeMinardi]

A good explanation. I liked the way the rotation in 2D is used to explain the transformation from one orientation to another. He could have dwelled on the final solution at 10:40 a little bit to let the viewer soak it in ( and not hide it with his hand). I would also suggest holding a physical object at the start rather than waving hands so much to reduce the level of abstraction necessary to visualize. As a 3D CAD guy I use quats quite a bit as they do not suffer from gimbal lock as found with homogeneous transformation matrices. That factor, and the speed of execution compared to traditional matrix manipulation, make them the preferred controllers for 3D animation.

[@streak1burntrubber]

Real numbers... stupid name. Imaginary numbers... stupid name... Complex numbers... stupid name........... We need to get better people to come up with names for this stuff.

[@harry_page]

How about direct numbers instead of real numbers and lateral numbers instead of imaginary numbers? Those names are what Gauss came up with I think

[@sebastiangudino9377]

Dude, Hypercomplex Number (The superset of quaternions) is a really cool name!

[@mydogskips2]

+Tsavorite Prince I like that, does it include 3D numbers, or is it just 2D? Isn't a plane only two dimensional(and thus wouldn't a planar number define only a 2D space)? I mean, isn't a complex number just a number with a real component and an imaginary component? So this wouldn't necessarily mean it's a 3D number. Wouldn't a third component/coordinate be needed?

[@AttilaAsztalos]

Careful there. That's how you get Strange quarks and Charm quarks...

[@jeerdace8625]

Tsavorite Prince Don't imaginary numbers fall under the category of complex numbers? Therefore, bilinear numbers would be planar numbers right? Or have I hit my head against a wall?

[@woobmonkey]

First, new SingingBanana vids, now Dr. Grime on Numberphile! Quite the week for the most awesome maths popularizer on the internet!

[@D4v1ks]

Great video.I always enjoy to understand why something works, instead of just using it.Thanks

[@clawpuss2]

Love the infectious enthusiasm of this guy.

[@frank39pw]

Grime's explanation was, far and away, the best I've ever seen, or read, on quaternions. I could parrot the multiplications but had little feel for the What's and the Why's.

[@RickusDomesticus]

This is by far the simplest explanation I have ever watch

[@veggiet2009]

You have no idea how happy it makes me to see Numberphile cover quaternions. What I would like to know about and maybe Dr. Grime knows about the Rotors in Geometric Algrebra. Like Quaternions, but rotors can be construction to do rotations in N dimensions. I would very much like to figure out how to write in code a 6d rotor, which if I'm remembering correctly is required to rotate a 4 Dimensional construction. I have read a bit of documentation and actually a piece of code, but I still do not grasp the concept.

[@timothycalco8089]

This guy’s excitement is contagious!

[@naimulhaq9626]

Absolutely beautiful presentation. Thank you Dr. Grime.

[@onurereren]

I supposedly learned about quaternions when I studied astronautical engineering, but it was told neither this clearly nor with this much enthusiasm. At last, after a decade and a half, I understand where they come from! :)

[@oken8501]

Wow, you made me understand what are complex numbers for... something no math teacher ever managed to do or actually even tried before.

[@Arsbars1]

great content!!! Amazing! Idk maybe some have already pointed out, at 4:46 the animation got it wrong , at 45 degrees you rotate by 45 again , you end up at i ( 90 degrees, or at 1 on Y axis)

[@beat461]

this guy is always so full of enthusiasm

[@redouanekachach9181]

Simple and straightforward explanation. Thanks Sir.

[@ma888u]

Dear Sir... you are ingenious and from now on i call myself your groupie of science!!! :-) :D You have to be a prodigy to explain something so complex so simple!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[@kinanali2668]

U can't imagine how much u helped me through this video Thanks

[@enderuslu725]

What I like about western education is they give its history and the philosophical background of the concepts and theories so that you have reason to grasp them better. It is the best simple explanation including the complex numbers I have ever heard.

[@cts3md]

Awesome video very clear and well presented! Thank you so much for posting!

[@PaltryPete]

I like how you shoot your videos on various different locations. It does something, I think.

[@castromanist]

In the last days I've fallen in love with this channel...