Fantastic Quaternions - Numberphile

Dr James Grime discusses a type of number beyond the complex numbers, and why they are useful.
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[rudymartin]

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.

[AlanKey]

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.

[pegy]

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.

[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!

[gabrielkwiecinskiantunes]

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

[skroot]

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!

[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.

[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...

[fakjbf]

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.

[komrad]

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

[brianhoskins]

Numberphile is easily the best channel on all of YouTube.

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.

[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 >.>

[Entropyko]

"lots of i" love that Brit-speak.

You explained it very well James!

[michaelgodfrey]

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

[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.

[Cerealbox]

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.

[theatheistpaladin]

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

[TecaLucasndChannel]

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

[Madsy]

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?