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Quaternions and spherical trigonometry

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Re: Quaternions and spherical trigonometry

#21

Post by quantadev »

wanderingmoose wrote: post #18
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…

I really like the welch labs series on imaginary numbers which covers the first part of what you talk about -- leveling up the notion of what a complex number is. Though his focus was more on solving simple equations with no real roots, but really detailing how/what is really going on. It is a great precursor to then thinking about quaternions https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX...

That exact video is the one that always comes to my mind when thinking of YT videos on this! I've seen it years ago. Definitely worth a watch for anyone who hasn't see it!

Re: Quaternions and spherical trigonometry

#22

Post by Sniffnoy »

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…

Octonions and so on up are indeed a thing, but I don't think they do what you want. Even aside from the fact that they're restricted to power-of-2 dimensions, their algebraic properties get worse as you iterate the Cayley-Dickson process. The octonions aren't even an associative algebra, although they do have some weaker associativity properties which I'll skip detailing here. The quaternions are as far as most mathematicians are willing to go -- non-commutativity is commonplace, but who wants to deal with non-associativity?

But while the octonions at least have some mathematical relevance (they're actually connected to various exceptional objects, such as the exception Lie group G_2!), the sedenions and beyond basically don't. They have a tiny bit of associativity but not enough that they connect to any things or that hardly anyone wants to study them -- and worse yet, there are zero divisors so cancellation (ab=ac => b=c for nonzero a) doesn't even hold. (Inverses exist, yes, but without associativity, inverses don't imply cancellation! And therefore aren't much use.)

As another commenter mentioned, what you might be looking for instead if it's orthogonality you're focused on is Clifford algebras (aka geometric algebra). However, if you want to get the complex numbers or quaternions out of it, you'd need to use a negative-definite quadratic form -- if you use a positive-definite one, you'd instead get the split-complex numbers, which are much less interesting (and you'd get something similar instead of the quaternions).

Re: Quaternions and spherical trigonometry

#23

Post by quantadev »

AnIrishDuck wrote: post #19
hgomersall wrote: post #17

Earlier quoted context omitted.

Geometric algebra would be what you're looking for here. This is a great intro to the topic: https://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-ar...

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 Space, and so in some sense we can treat this space as if it's real geometry. I know a TON of research is done in this space, especially around what they call 'Mechanistic Interpretability' of LLMs.

Re: Quaternions and spherical trigonometry

#24

Post by alfiedotwtf »

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 Algebra isn’t it? Oh… dont tell me Quantum Spin just called Spin because it’s a Spinor rather than something actually metaphorically spinning?!

Please chime in if you know what I’m talking about and can confirm this or shoot it down.

Re: Quaternions and spherical trigonometry

#25

Post by nh23423fefe »

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…

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.

Re: Quaternions and spherical trigonometry

#26

Post by jdranczewski »

incognito124 wrote: post #3

Related: https://eater.net/quaternions

This was an invaluable resource 5 years ago when I was working on a summer research project making a ray tracing-based optical levitation simulator - initially it felt a bit insane to try to deeply understand this obscure bit of maths to implement rotations, but once it clicked it clicked. Quaternions ended up being a super neat formalism for writing and computing rotational equations of motion.

https://github.com/jdranczewski/optical-levitation-raytracin... for my repo, and https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5165 for the rotational dynamics with quaternions.

Re: Quaternions and spherical trigonometry

#27

Post by particleguy »

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…

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

Re: Quaternions and spherical trigonometry

#28

Post by alfiedotwtf »

particleguy wrote: post #27
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…

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 don’t know what’s changed (maybe maturity or maybe Vyvanse lol) but I’m slowly putting the pieces together. It’s always been in my outer periphery but still out of reach. Your confirmation has and will change my life. Maybe not career wise or life altering seen from the outside, but hot damn you have at least cleared my constant nagging guilt for not perusing maths and physics because you’ve just made it slightly closer within reach. Can’t wait for the book to arrive. Thank you!!!

Re: Quaternions and spherical trigonometry

#29

Post by aap_ »

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…

To understand spin it's good to consider a gyroscope. when it has a lot of angular momentum there are two stable states for it in a gravitational field: aligned or anti-aligned with gravitation, up or down. in all other cases the gyroscope precesses. a spinor doesn't spin quite like a gyroscope but it is spinning in a sense (after all spin is angular momentum). but just like the gyroscope, you can think of it as having two stable states: in alignment with a magnetic field, or in anti-alignment. and because the magnetic field is a measure of some kind of rotation it can add to or subtract from the angular momentum of a spinor. this difference is "felt" as a negative or positive potential difference. this you can think of as two opposite forces on the spinor that split it apart into up and down components. the interesting thing is that a spinor with an arbitrary axis can always be written as the sum/superposition of an up and down spinor for some chosen direction. turns out quaternions have precisely the properties that you need to model this. i hope this was intelligible, it's a bit hard to put the geometry into words.

Re: Quaternions and spherical trigonometry

#30

Post by markisus »

nh23423fefe wrote: post #25
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…

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.
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