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Quaternions and spherical trigonometry

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Re: Quaternions and spherical trigonometry

#41

Post by raphlinus »

alfiedotwtf wrote: post #28
particleguy wrote: post #27

Earlier quoted context omitted.

Yes, Quantum Field Theory can be explained through Lie groups. SU(2) is isomorphic to the quaternions of norm 1, and SU(2) is important if you want to understand the Lorentz group and Poincare group, which represent the symmetries of spacetime and special relativity. Check out the text book Physics From Symmetry by Jakob Schwichtenberg if you would like an approach that derives modern physics primarily from algebra

OMG thank you. Purchasing right now! You DO NOT understand how happy I am right now. Truely! I did general physics for a year at uni as part of my Computer Engineering course, then switching to Computer Science where I picked up a year of quantum mechanics. Since then whenever I lay in bed and thought about physics I would end up awake for hours. So damn interesting but the maths always held me back, so sadly gave up…

I just want to say I'm rooting for you, and hope you enjoy the book and learn a lot from it.

I had a bad experience with complex analysis as a teen (took a grad class that was a bit over my head). Many years later, I got Tristan Needham's "Visual Complex Analysis" and the whole thing clicked for me - I'm a visual person and do a lot of geometry. I hope your experience is similar.

Re: Quaternions and spherical trigonometry

#42

Post by AnIrishDuck »

quantadev wrote: post #23
AnIrishDuck wrote: post #19

Earlier quoted context omitted.

Also the (provocatively titled) "Let's Remove Quaternions from every 3d Engine" [1] Spoiler alert: rotors are mechanically identical to quaternions, while being easier to understand. If you understand rotors, you understand quaternions. You can fit the laws you need to understand rotors on a business card. Plus, rotors abstract to higher and lower (well, there's only one plane and its two respective orientations in 2…

I had never even heard of rotors! Thanks for this. I watched that video. The video doesn't really explain how it extends to higher dimensions tho, that I could discern. I wonder how/if any of this can be applied to LLMs 'Semantic Space'. As you might know, Vector Databases are used a lot (especially with RAG - Retrieval Augmented Generation) mainly for Cosine Similarity, but there is a 'directionality' in Semantic Sp…

> The video doesn't really explain how it extends to higher dimensions tho, that I could discern.

The neat thing is that it "extends" automatically. The math is exactly the same. You literally just apply the same fundamental rules with an additional basis vector and it all just works.

MacDonald's book [1] proves this more formally. Another neat thing is there are two ways to prove it. The first is the geometric two-reflections-is-a-rotation trick given in the linked article. The second is straightforward algebraic manipulation of terms via properties of the geometric product. It's in the book and I can try to regurgitate it here if there's interest; I personally found this formulation easier to follow.

If you really want your mind blown, look into the GA formulation of Maxwell's laws and the associated extension to the spacetime (4d) algebra, which actually makes them simpler. That's derived in MacDonald's book on "Geometric Calculus" [2]. There's all kinds of other cool ideas in that book like a GA formulation of the fundamental law of calculus from which you can derive a lot of the "lesser" theorems like Green's law.

Take all of this with a grain of salt. I'm merely an enthusiast and fan, not an expert. And GA unfortunately has (from what I can tell) some standardization and nomenclature issues (e.g. disagreement over the true "dot product" among various similar but technically distinct formulations)

> I wonder how/if any of this can be applied to LLMs 'Semantic Space'.

Yeah, an interesting point. Geometric and linear algebra are two sides of the same coin; there's a reason why MacDonald's first book is called _Linear and_ Geometric Algebra. In that sense, Geometric Algebra is another way of looking at common Linear Algebra concepts where algebraic operations often have a sensible geometric meaning.

1. https://www.faculty.luther.edu/~macdonal/laga/ 2. https://www.faculty.luther.edu/~macdonal/vagc/

Re: Quaternions and spherical trigonometry

#43

Post by immibis »

quantadev wrote: post #16

Just a tiny rant: In my view complex numbers are really about the concept of "orthogonality". The complex 'dimension' is orthogonal to the 'real' dimension, but anything in reality that's a continuum of values can be seen as a dimension, and therefore each one must have an orthogonal. That is, whenever you have a direction in a higher dimensional space (regardless of dimensionality) any vector will have a normal dire…

You can just start with N orthogonal directions which is R^N. There's no need to use the Cayley-Dickson construction to get orthogonality.

Re: Quaternions and spherical trigonometry

#44

Post by pjbk »

You can certainly use quaternions (as Hamilton demonstrated) or a Clifford algebra to recover the spherical trigonometry laws, but plain vectors work too in a short derivation. It is actually one of the simple exercises introducing reciprocal bases in Louis Brand's book Vector and Tensor Analysis (https://archive.org/details/vectortensoranal00branrich) or its abridged version, Vector Calculus.

Re: Quaternions and spherical trigonometry

#45

Post by pjbk »

quantadev wrote: post #16

Just a tiny rant: In my view complex numbers are really about the concept of "orthogonality". The complex 'dimension' is orthogonal to the 'real' dimension, but anything in reality that's a continuum of values can be seen as a dimension, and therefore each one must have an orthogonal. That is, whenever you have a direction in a higher dimensional space (regardless of dimensionality) any vector will have a normal dire…

Algebraically, and also in group theory, it's exactly that. A complex plane is re-interpreted as a half-way mirror operation on an orthogonal axis or quadrature plane (eg the imaginary axis in the case of the 2D Argand plane). Which is what you you get when you multiply by i in complex numbers or i/j/k in quaternions - a 90 degree rotation. The correct way to do arbitrary rotations in this case is to use an exponential process, and then you get Euler's equation, the quaternion symmetry operation or, in general, the exponential map of an infinitesimal transformation in a Lie group.

In higher dimensions you get other type of (sometimes weird) operations, related to the Cartan–Dieudonné theorem.

Re: Quaternions and spherical trigonometry

#46

Post by alfiedotwtf »

raphlinus wrote: post #41
alfiedotwtf wrote: post #28

Earlier quoted context omitted.

OMG thank you. Purchasing right now! You DO NOT understand how happy I am right now. Truely! I did general physics for a year at uni as part of my Computer Engineering course, then switching to Computer Science where I picked up a year of quantum mechanics. Since then whenever I lay in bed and thought about physics I would end up awake for hours. So damn interesting but the maths always held me back, so sadly gave up…

I just want to say I'm rooting for you, and hope you enjoy the book and learn a lot from it. I had a bad experience with complex analysis as a teen (took a grad class that was a bit over my head). Many years later, I got Tristan Needham's "Visual Complex Analysis" and the whole thing clicked for me - I'm a visual person and do a lot of geometry. I hope your experience is similar.

haha thank you!

Awesome :) Yes, I saw that book this morning on my Amazon travels... though I'll put it in my wishlist because I think this morning blew out my yearly book allocation lol.

Re: Quaternions and spherical trigonometry

#47

Post by alfiedotwtf »

jeffwass wrote: post #39
alfiedotwtf wrote: post #24

Hold up! Omg, can someone who’s done physics chime in please… whenever I’ve looked at GUT etc, I’ve always seen U(n), SU(n), but never knew what they were - are they what’s referred to in this article? Is that just the Unitary Group and Special Unitary Group??! All that time I thought it was all impenetrable but it’s just algebra? Omg wow... the theoretical physics I’m talking about is just quaternions and Lie Algebr…

Quantum spin is an intrinsic angular momentum of a particle. It's angular momentum that is 'just there' as a key component of the particle. In early days it was hypothesised that particles were spinning about their own axes, but this isn't accurate. All the interesting stuff of Spin from its quantizable nature, the non-commutatability of spin measurements along orthogonal directions, the very different fundamental be…

I think we maybe saying the same thing (unless I'm not reading that right) - Spin is not named because it's physically spinning (ok intrinsically perhaps, but that still isn't intuitive to me) but because the way we measure its interactions it's easier to describe using Spinors?

Re: Quaternions and spherical trigonometry

#48

Post by alfiedotwtf »

JadeNB wrote: post #38
alfiedotwtf wrote: post #24

Hold up! Omg, can someone who’s done physics chime in please… whenever I’ve looked at GUT etc, I’ve always seen U(n), SU(n), but never knew what they were - are they what’s referred to in this article? Is that just the Unitary Group and Special Unitary Group??! All that time I thought it was all impenetrable but it’s just algebra? Omg wow... the theoretical physics I’m talking about is just quaternions and Lie Algebr…

> Oh… dont tell me Quantum Spin just called Spin because it’s a Spinor rather than something actually metaphorically spinning?! That's a great way to understand them if you are already comfortable with the algebra of spinors and spin groups, but it doesn't short-circuit the history—spinors were so called after quantum spin ( https://en.wikipedia.org/wiki/Spinor#History ), and I believe that was so called because, yes…

Interesting.

I've tried many times to go through Modern Algebra texts, and so I on-the-surface get that it's an algebra of sorts.

When I have enough time I'm going to finally go through mathacademy.com, because I think it really does suck not knowing advanced maths but want to do the hard sciences

Re: Quaternions and spherical trigonometry

#49

Post by quantadev »

AnIrishDuck wrote: post #42
quantadev wrote: post #23

Earlier quoted context omitted.

I had never even heard of rotors! Thanks for this. I watched that video. The video doesn't really explain how it extends to higher dimensions tho, that I could discern. I wonder how/if any of this can be applied to LLMs 'Semantic Space'. As you might know, Vector Databases are used a lot (especially with RAG - Retrieval Augmented Generation) mainly for Cosine Similarity, but there is a 'directionality' in Semantic Sp…

> The video doesn't really explain how it extends to higher dimensions tho, that I could discern. The neat thing is that it "extends" automatically. The math is exactly the same. You literally just apply the same fundamental rules with an additional basis vector and it all just works. MacDonald's book [1] proves this more formally. Another neat thing is there are two ways to prove it. The first is the geometric two-r…

Interesting ideas there thanks. I do know about that Maxwell derivation that involves Minkowski space, Lorentz transform consistency, etc, although I haven't fully memorized how it works, so that I can conjure up how it works from memory. I don't really think in equations, I think in visualizations, so I know a lot more than I can prove with math. You're right it's mind-blowing stuff for people like us that are interested in it.

Re: Quaternions and spherical trigonometry

#50

Post by btilly »

markisus wrote: post #30
nh23423fefe wrote: post #25

Earlier quoted context omitted.

I don't agree. Complex numbers are the algebraic closure of the reals. Or the quotient of the real polynomial ring by (x^2+1=0). Or whatever other construction. The multiplication rule is the essence of C. Orthogonality is captured linear algebra over R^2, but R^2 isn't a field or an algebra.

I think there is still a geometric viewpoint you can bring to the multiplicative structure of C. For example there is the extremely natural homeomorphism between unit C and SO(2). And C minus origin to (R+, SO(2)). It’s completely intuitive for mathematicians to say that 1 and i are separated by 90 degrees.

Indeed, there is a polar coordinates representation of the complex numbers with the nice property that when you multiply you multiply lengths and add angles.
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