a quaternion version of Euler's formula
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- Опубликовано: 25 июл 2024
- WHAT IS THIS? INK? INK?! SINCE WHEN ARE YOU INTO...UGH...INK? OH MY GOODNESS HOW COUL-
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Link to short referenced in video: • #quaternion exponents ...
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I think you should read the description.
whatthefuckishappening?
'-'
I did. I'm still processing it
😂😂😂😂🤣🤣🤣👏👏
I really liked this video because of all the math this dude did. What a guy.
I am waiting to see the octonions video with the nonassocietivity hell 😂
Fascinating, especially the last result which I hadn't seen before.
I find that an easy way to grasp quaternionic behavior under exponentiation is to see that any single quaternion (other than a purely real one) defines a complex sub-algebra of the quaternions and that sub-algebra behaves the same way as the complex numbers; its purely imaginary part, unitized, plays the same role as the complex number i. A lot of unintuitive stuff got much simpler when I understood that.
What a gnarly way to start the week.
Thank you, professor.
This is the first time quaternions made any sense to me, in spite of several attemps. As ususal presented in a clear no-nonsense way. Thanks a lot!❤
haha same this makes alot more sense then anything ive seen
wow, never though that i^j would be equal to i*j=k, thats is cool. Nice video as always Michael! :D
Well, ofc you wouldn't think that because it's BY DEFINITION. It's one of the things that make quaternions... quaternions! 😉
Just like i^2 = i*i = -1 is by definition what makes a complex number complex.
@@mikeiavelli The definition says i*j=k, but it's not immediately obvious from the definition that i^j=k, at least not to me.
I really liked this video because of all the math this dude did. what a guy
Both of those formulae are special cases of the Clifford algebra formulation of 3D rotations :)
Lovely video, as usual. Great channel to watch. Cheers.
Fun video. BTW, the matrix representation of the quaternions is related to the groups SU(2) and its covering group SO(3), and thus they can be used to represent rotations in three dimensions.
Nice connection. Thanks!
We're nudging closer to the Clifford Algebra and Geometric Algebra
This was a very interesting, fun presentation. Thank you professor))
Very interesting and beautiful results. Thanks for sharing.
Fun video. Lord Kelvin declared that quaternions, “though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell” 👿
8:46 almost fell over but kept the balance. well done professor
I really loved the video it's very good . Thank you sir😊 we will support you
God I love mathematics. It's so lovely to play with ideas in this way.
when I learned that Pauli matrices have such a similar Euler's identity I was really surprised. But if you add the identity you get exactly the quaternion algebra, so knowing that, the quaternion exponentiation makes perfect sense
Wonderful! Thanks so much 😮
I loved this! Would love to see this extended to the Clifford representation.
Whenever I see a video about complex numbers I automatically wonder about quaternions. Whenever there’s quaternions I automatically wonder about geometric algebras.
Fun fact about Euclidean (Clifford) geometric algebra is that the basis vectors square to 1. Products of non-parallel vectors create bivectors. Unit bivectors square to -1 so they act like i in Euler’s formula. In fact, the scalars plus bivectors form a sub algebra that’s equivalent to the quaternions!
@@zemoxian Even when the vectors have more than 3 dimensions?
if you feed vectors into it you get hyperbolic rotations
@@Apollorion clifford/geometric algebra works independent of the dimension of the vector space that you feed it. It just requires a vector space *and* the inner product structure you feed it (i.e. if some of the vectors square to -1, or even 0).
Even a generalization of the cross product called the 'wedge product' works in a geometric algebra generated by a vector space of more than 3 dimensions. It comes down to the way the cross working based off what the orthogonal complement *is* of the vector you're feeding it (and the cross product fails, because it demands to output just a vector while there are more than 1 orthogonal vectors now to a given vector in 4D. The wedge product which works on multivectors doesn't have this limitation)
EDIT: One more thing I want to add is a lot of people will mention wedge-products as not having a visualization to them. I would be wary of statements like these, they can be represented as the plane spanned by those two vectors you're 'wedging' (but this plane, or circle, or whatever also has an orientation to it. So e1 ^ e2 has the opposite orientation to e2 ^ e1). They may have very good algebraic explanations, but not the geometric explanation handy, and that's totally ok! (the vice-versa also can happen, and you should be wary of that too when/if you run into it).
@@monadic_monastic69 Thank you for your response. I realise I do not know enough about the clifford/geometric algebra yet.
I really liked this video because of all the math this dude did. what a guy :)
regarding the end, there: it feels as though we'd need to define a separate left-sided exponentiation to account for the lack of commutativity, at a bare minimum
The title for this video is quite clear and descriptive. The title card is also quite clear and descriptive. Thank you.
Great explanation of a tough topic. Great Job!
This is exactly what I needed!!
The reason we use H for the quarternions is only secondarily because the letter H stands for Hamilton.
Firstly: we can't use the now obvs is Q because that's already taken.
This is, of course, a rational reason.
Exponentiation of a matrix is just applying the series representation of the exponential to the matrix. It is so amazing to see the concept of exponentials generalize this way... Same for the logarithm. Thank you for making the video!!
Yes. My favorite example of generalized exponentiation is the math of squeezed light, such as what LIGO uses. It's the sinh and cosh of the sum/ difference of the creation and annihilation operators, the quantum mechanical operators that mean "add a particle" and "remove one". Since sinh and cosh are just sums of exponentials, we are exponentiating to the power of "add a particle" etc and it actually works!
@@Tehom1this sounds awesome, is there a RUclips resource you can point to??
@@scalex1882 I read this in a scientific paper and I'm afraid I don't know of a youtube video about it, sorry.
It is not clear to me why i^j should be k instead of -k. Why do we say that (e^(i*pi/2))^j is e^(ij*pi/2) instead of e^(ji*pi/2)?
Because multiplication is not commutative it is anti commutative that is just (iπ/2)j that is ijπ/2 but if we were to write jiπ/2 we need to add a -ve signed as well so the exponent can be written as -jiπ/2
Consistency with the order of the imaginary and its coefficient might be a good idea, especially when you start getting into quaternions.
I mean, if you try to go the route of complexifying a complex number, (a + bi) + (c + di)j = a + bi + cj + dk is fine but (a + ib) + j(c + id) = a + ib + jc - kd is slightly problematic.
You could do (a + ib) + j(c - id) = a + ib + jc + kd, of course, which is a nice callback to the particular complex matrix form you use; though I prefer a different form.
Anyway, the signatures are pretty arbitrary here (as sign is, generally) so long as you are consistent, throughout.
I'd just stick with trailing imaginaries, personally, but then you run into a formatting issue with Euler's formula. If you want to use the leading i in Euler's and the conjugate imaginary component in the matrix it's probably best to stick with leading everywhere to avoid problems.
Really interesting, thanks!
As a physicist I feel like you should discuss the connection to SU(2). It's quite simple to show the version of Euler's formula in that context as well.
My first introduction to quaternions was glancing through an applied electrical engineering text. Fascinating something so theoretical is so essential to describe electrical behavior.
The unit quaternion algebra is isomorphic to su(2), which is the Lie algebra that describes the angular momentum of fermionic particles (in other words, the so-called 1/2 spin of the elementary matter particles such as the electron, quark, and neutrino). And the "vector algebra" (dot product and cross product) which is used to write the classical laws of EM were historically extracted from the quaternion algebra to begin with. It is indeed fascinating.
I was introduced to quaternions by the video done by 3Blue1Brown... and they keep getting stranger and more interesting with every new thing I learn about them. The exponential transformations exposed in this video are an especially brain-bending result.
I'd suggest looking into the clifford/geometric algebra way of looking at quaternions next. They'll certainly get more interesting, but less strange (an algebra on planes, and there are three planes in 3D space: xy, yz, zx. Oh btw, i, j, k, are basically the normal vectors to these guys if you wanted to think of it that way), and though I like 99.9999% of 3blue1brown's vids, I don't agree with the framing of quaternions as inherently "4-dimensional creatures" (and the clifford/geometric algebra view on this clears this up, also clears up what exponentials mean when you're putting in different objects in there).
While at the same time, if you *want* 4D things, you can look at the clifford/geometric algebra generated by a 4D vector space: 'space-time algebra'.
@@monadic_monastic69 Unless the number of videos produced by 3blue1brown is a multiple of a million, the percentage of them you like cannot be 99.9999.
I actually finally understand quaterions now.
13:45 ... My heart sank as I expected you to say " and this would be a nice place to stop" I watched to the end and to be honest, didn't need the matrix stuff, I was already knocked out by the beauty of it all. I dare some elerctrickery wiz will find some use for 'i' and 'k' in their modelling.
I love the way you present the ideas. Engaging and clear.
You asked about the red-brown chalk once. It is a bit hard to see, for example at 2:06. around "example calculations". But these are quibbles. It works for boxes and lines, and the rest of the boards are clear to follow.
your feelings are irrational
Very very interesting video.Million thanks from Morocco.
thanks for your persistence
There's a Quarternion version? Damn, you learn something new everyday.
When teaching the addition of many disparate components, I like to underline the components as I deal with them.
This establishes a bookkeeping methodology for the students (and me!) that prevents losing track of terms in a long list.
Some years ago I calculated a general formula for a quaternion raised to a quaternion power. Quite messy and arduos, but was fun and the result is quite neat, a scaled rotation along the axis of both original quations followed by a rotation about the axis of their cross product. (Everytime I say "the axis of" I mean the axis defined by the purely imaginary part, and the cross product being within the imaginary parts also)
Edit: of course, this was all abusing notation and glossing over the sketchiness of using exponentials on a non-commutative algebra, but I was on my first year of college so chill I didn't have the knowledge to take that much into consideration xD
In a way, e^{i\theta} is an abuse of notation.
@@cxpKSip notations are made to be abused stretched to the ultimate - that's how all ideas are tested and great ones are created. Keep it up!
What is this video description lol
Great vid, I love seeing quaternions at the zoo like this. Wish I saw more in the wild...maybe.
Quaternions written as matrices were the theme of French most difficult math contest (X-ENS, Maths A) last week, to enter the best engineering school.
interesting ! thank you very much
What are the connection between the quaternions and the Pauli matrices?
The Pauli matrices are generators of the Lie group of the unitary 2x2 matrices with determinant +1, i.e. SU(2). The key relation is
U = exp(b sigma_x + c sigma_y + d sigma_z)
in which
sigma_x = [0 1]
[1 0]
sigma_y = [0 -i]
[i 0]
sigma_z = [1 0]
[0 -1]
(i.e. the Pauli matrices)
b, c and d are real coefficients
U is an arbitrary member of SU(2)
The relation to quaternions is by putting
sigma_x = i, sigma_y = j, sigma_z = k
U = q, an arbitrary quaternion of modulus 1
Michael's result for a=0 is
q = exp(bi + cj + dk) = cos(B) + I sin(B)
with I = (bi + cj + dk)/B, B = sqrt(b^2+c^2+d^2)
That description though lol
that was really helpful
I did not expect spinors to make a surprise appearance at the end. But you are spinning things so I guess I should've
More Quaternions!
I think the contradiction showed in the short video can also be seen here.
when you take i^j=(e^{i*pi/2})^j pulling j inside of the exponential isn't well defined I think, would it necessarily be multiplication from the left or should it be from the right? because of the commutation relations i and j have it can be either e^{k*pi/2} or e^{-k*pi/2}.
also, as usual great video- I love those videos of yours where you push poke known math in unusual ways to see what comes out
for the exercise at 5:27, writing q for a + bi + cj + dk and q*/|q|^2 for (a - bi - cj - dk)/(a^2 + b^2 + c^2 + d^2), one needs to check _both_ q(q*/|q|^2) = 1 and (q*/|q|^2)q = 1, since quaternions are noncommutative - we have |q| ≠ 0 if and only if q ≠ 0 so q*/|q|^2 gives the inverse of any nonzero quaternion, and one can clear the denominators in the checks, meaning it is equivalent to check that qq* = |q|^2 = q*q.
You do not really need to look at both if you find that one of them is |q|². If you found that qq*=|q|², you could note that q*q=(qq*)*=(|q|²)*=|q|², and say that you found the answer since |q|² is a real number.
If you would find that q*q≠|q|², this would be a non-issue since therefore q^(-1)≠q*/|q|² as noted above.
I bearly understand normal math and then there is this
Professor Penn,
Why do I think you are a genius?
18:45 Why is it log(A)*B rather than e.g. B*log(A)? Isn't that rather arbitrary choice (unless the matrices commute)?
The quatirion field post in exam MathA 2023 in France is beautiful exam to more understanding the quaternions field specialy her matrix represtation
i^j = ±k depending on how you apply rule for the exponential of an exponential (right vs left).
This is true of matrix exponentiation as well-it gives rise to two kinds of exponentiation-left and right-which for commutative multiplication are the same. What is less obvious to me is what the principal branch of the quaternion logarithm should be. If you have the Log of -1, which quaternion do you use? Any purely imaginary quaternion with modulus 1 will work.
Neat!
Hi Michael, there is a major simplification when applying the Power Rule for Exponents to the quaternions, you said exp(i * pi /2) ^ j is exp(i * j * pi / 2) yet I think j being on the right side of i is not so obvious. Why is this the case? Since normally this rule was applied to commutative numbers, now with quaternions there must be a special explanation of this specific order.
Edit: Probably this is related to the problem you mention in the short video.
i suggest you start with q=a+b1*i+b2*j+b3*k and b=sqrt(b1^2+b2^2+b3^2) then q=a+b*Î with î the specific normalized quarternion here with |î|=1 and î=b1/b*i+b2/b*j+b3/b*k, then e^q becomes very simular to the complex case: q=a+b*î and e^q=e^a*(cos(b)+sin(b)*î) which notation seems much more alike that in the complex situation. Right?
nicenice nice i love quaternions :o
1:09 The 'where' could/should include ijk = -1. Then the 6 relations below that follow from these 4 relations, which are easier to remember.
THE DESCRIPTION LMAO
How much of Complex analysis can be extended to the Quaternions? I have often wondered about this.
Do you have any sources or references?
Using i as quaternion and complex in same equation is VERY confusing.
It's somewhat related to multivariate complex analysis. But I'm not an expert and I don't know how the loss of commutativity changes the analysis.
Not much, because they are non commutative. For example, the Foundamental Theorem of Algebra fails!
@@michaelaristidou2605 Thanks.
The analogue of complex analysis for quaternions is called quaternionic analysis.
@@schweinmachtbree1013 Is it good for anything?
I guess the description is a mathematical secret code
Nice can you do maxwells equations next?!
I really like this video this guy is awesome CHALK CHALK CHALK BLACK BOARD
Suddenly I feel thankful for Heaviside's version of Maxwell's equations
Quaternions' didn't click for me until I learned a little geometric algebra.
Wonderful video but I haven't understood the last step, could you please explain or prove it .
I would love to thank you a lot for the video.
I have a question (extremely curious):
Is that method applicable for the higher Hypercomplex Numbers such as Octonions and Sedenions or there will be some differences in the "B" and "I" construction?
Generally, every time you go up to a 'next level' of complexity, the algebra changes; in quaternions for instance, commutativity of components under multiplication is lost. I haven't played with Octonions yet, but expect it's a different game.
Plotting the unit circle we see that the form given in this video is related to the identity cos squared plus sin squared = 1 by multiplying by the complex conjugate
Gently press
Is this an example of the exponential map from a Lie algebra to its Lie group? The group being the multiplicative group of quaternions of magnitude 1 (so-called versors, isomorphic to SU(2)) and the algebra being the infinitesimal displacements from the identity element, q=1.
I'd be grateful for a reply though I'm not sure if I'd understand it -- despite Michael's efforts, Lie algebras still turn my brain to jelly!
Yes, this is correct.
Well, it doesn't commute, I was waiting for that...
It's linear transformations, I use those tools a lot for 3D cg
interesting that the complex numbers can't just be treated as a plane within the quarternions, but I guess that kind of makes sense the same way that imaginary numbers can't be treated like a number line within the complex numbers
13:41 this is in fact the quaternionic version of this formula
Quaternions, octonions, tetrads, tensors, twistors and vectors are devices for physical description of movement and variation of quantities.
This is STRICT computer graphics. Hamilton among many other mathematicians are geniuses.
Question: can you take the logarithm of a matrix by using the spectral decomposition? I've never seen that done but it seems to me that if you can use the spectral decomposition to exponentiate a matrix then it should also be possible to take a matrix and use the spectral decomposition to take its logarithm.
Equals evil in biome
You can likewise show that j^i = 1/k, which is pretty cool.
Editor note: @5:26 - don't use red chalk for text, just outline. Can't read "exercise". Also, it seems there is a mild glare in the center of your board. Change to position lights or camera to not glare.
Wait so I know i = e^i * pi/2 and j and k but what aboit a general quaternion like i+2j+k? What is that in exponential form?
Seems like people are very interested in algebraic structures
This comes from Clifford algebra, or as it's known these days, geometric algebra.
Any chance of getting a link to the video you're talking about that explains why this definition is sketchy? :)
it'll be posted tomorrow :)
-Stephanie
MP Editor
This is quite an intuitive reconstruction of Euler’s formula. But after watching, I’m left to wonder: what about infinite tetrations of quaternions? Are there any that converge to a real and/or transcendental number? And what about the reciprocal of an infinite tetration of quaternions?
Do you know the Dihedrons? ruclips.net/video/lqH4BLHGsFw/видео.html
What should be the Euler formula for them?
Can jAe^jB((iCe^iD) - (iEe^iF)) be put in the form I*Ge^I*H, where i is complex, j is part of the quaternions, and I is the same as your notation?
I came up with a better, simpler and easier to understand implementation of Euler's formula in quaternion form that can be implemented directly into a computer program to do rotations about an arbitrary axis in 3D space. @FractalWoman "Demystifying Sir William Rowan Hamilton's Quaternions" ruclips.net/video/vXyNA0ORYfA/видео.htmlsi=62Eao0gdh7O5LPRp A link to the computer code is available in the description.
The ending is a bit concerning, as you mentioned.
We can still define exp(a+BꞮ) = e^a(cos(B)+Ɪsin(B)). From here, let's define the modulus of a quaternion q = a+bi+cj+dk as |q| = sqrt(a^2+b^2+c^2+d^2).
Then, letting q = a+bi+cj+dk = a+BꞮ as above, we can define a quaternionic logarithm as
log(q) = ln|q| + Ɪarccos(a/|q|), for some choice of arccos(a/|q|)
Then we can check that exp(log(q)) = q for all quaternions q.
Next, given two quaternions p and q, the next question is how to define q^p. The sneaky part here is that we have two reasonable options: exp(log(q)*p) or exp(p*log(q)). Note that these two do _not_ have to be equal to each other since quaternion multiplication is noncommutative.
My proposal, then, is to define two versions of exponentiation - right exponentiation and left exponentiation. At the risk of confusing with tetration, we use the following notations:
qᵖ = "q right exponentiated by p" = exp(log(q)*p)
ᵖq = "q left exponentiated by p" = exp(p*log(q))
So in your example of i raised to the j, we have to consider a few things:
For q = i, we have a = 0, B = 1, Ɪ = i, |q| = 1, so log(i) = ln(1)+i*arccos(0) = iπ/2, let's say.
iʲ = "i right exponentiated by j" = exp(log(i)*j) = exp(ijπ/2) = exp(kπ/2) = cos(π/2)+k*sin(π/2) = k
ʲi = "i left exponentiated by j" = exp(j*log(i)) = exp(jiπ/2) = exp(−kπ/2) = cos(−π/2)+k*sin(−π/2) = −k
This explains the discrepancy in your short where you talk about this.
Some interesting things to point out here: if r is a positive real number, then log(r) = ln|r|, which is a real number and commutes with all quaternions. So for any quaternion p and positive real number r, we have rᵖ = ᵖr (i.e., right and left exponentiation of r by q produces the same result). So we haven't introduced an incompatibility between e^p and exp(p) (using a good choice of argument).
As a bit of an addendum, in general, exp(p+q) will not be equal to exp(p)*exp(q) since quaternions don't commute. And certainly p^(q+s) will not be equal to p^q * p^s for quaternions p, q, and s (for either left or right exponentiation).
As a bit of fun, we can even consider
i¹⁺ʲ = exp(log(i)*(1+j)) = exp(iπ/2*(1+j)) = exp(iπ/2+kπ/2) = cos(π/√2) + i*√(2)sin(π/√2)/π + k*√(2)sin(π/√2)/π
whereas i¹*iʲ = i*k = −j
So these are very different results.
Does the entierty of the boards after 13:48 belong to :
e^q = e ^a ( ....
and there is no closing parenthesis.
Quaternions remind me a lot of the pseudo-Euclidean geometry of special relativity. One "real" component represents time, and three "imaginary" ones represent space.
Can we have nonstandard quaternions next, please?
Quaternion multiplication is order-dependent (not commutative): ij = -ji.
Does this mean that (a^b)(a^c) = a^(b+c) does not hold (even if a is real)?
Or is there some other reason that
k = ij = (e^i(pi/2))(e^j(pi/2)) = e^((i+j)(pi/2)) = e^((j+i)(pi/2)) = (e^j(pi/2))(e^i(pi/2)) = ji = -k
is not a valid derivation?
At 12:38 that seems to be to be a huge leap of faith. I expect that if you, say, expanded e^x to a series, stuck I in it, and simplified, I can see you could get the stated result, but saying it is so just because I^2=-1 seems like an awful stretch.
So what if you have an infinite expression of square root (i) with square root (j) inside of that with square root (k) inside of that and so on and then took the limit of that expression? Surely there’s a mathstack about that!
At the 19:42 mark, shouldn't the lower left "i" be "-i" ? In other words, shouldn't the lower left of your matrix "-c+di actually be -(c+di) = -c-di. When you do this, then you get the matrix [0 c+di], [-c-di 0] and when you let c=0 you get [0 i], [-i 0] = k. I can see this clearly when I use the 4x4 matrix implementation of quaternions directly which I show in my video: @FractalWoman "Demystifying Sir William Rowan Hamilton's Quaternions" ruclips.net/video/vXyNA0ORYfA/видео.htmlsi=62Eao0gdh7O5LPRp
I dont know why he said i^i is such as famous video concept. That has to be one of the easiest math results to arrive at that I can thing of...
how we can define quaternion j and k without any matrix ?
IN case of imaginary number i , i = square root of -1 but in quaternoins they use matrix but matrix is not a operation like square root , matrix is use for telling dimension of any thing , matrix is not a operation .
I wonder if we can use the properties of the exponential to write
exp(a+bi+cj+dk) = exp(a) exp(bi) exp(cj) exp(dk) = exp(a)(cos b + i sin b)(cos c + j sin c)(cos d + k sin d)
... but this looks very _sketchy_, given that i, j and k do NOT commute. Probably the right way to go about this path is to write the exp as a series and take proper care of the commutation relations.
You're right in that it's sketchy. In general, the power laws only actually apply when multiplication is commutative. You can pull out the scalar part since it commutes with the rest, but you can't split up the imaginary part.
It is definitely a wrong proof/derivation. The rigors generalized way to deal with something like that is using representation theory