One of perks of PhD of quantum physics is that quaternions get mundane, vide https://en.wikipedia.org/wiki/Pauli_matrices. SU(2) is everywhere (normalized quaternions). These are are more common than regular rotations of 3D space, O(3).
Because SU(2) we get a lot of interesting phenomena, including that there are two types of particles, bosons and fermions. We get some interesting phenomena that only rotating by 720deg (two full rotations) bring back to the initial state. And I am not talking only about USB-A, but about spinors (https://en.wikipedia.org/wiki/Spinor) - there are some party tricks around that (vide https://www.reddit.com/r/physicsmemes/comments/181oldw/a_ger...).
the late Doug Sweetser had in the 90s a website where he used quaternions (and quaternion analysis) for describing physics from his viewpoint. it was quit interesting. he had several github repositories [0] where he described his ideas. worth a look [0] https://github.com/dougsweetser?tab=repositories
I like quaternions as much as the next guy (I’ve used them in numerical computations etc), but what is it about them that makes them show up on the front page every few weeks?
In short:
scaling -> real numbers
1d rotations and scaling -> complex numbers
2d rotations and scaling -> quaternions
In the case of quaternions, there is called double-covering, which turns out (rather than being an artefact), play fundamental role in particle physics.
Baez wrote some ideas in [1], one I'm liking connects Lorentz group in dimensions 3,4,6 and 10 with the modular group SL(2,Z) that is at a crossroads of several hardcore math themes. For Lie algebras:
sl(2, R) ≅ so(2,1)
sl(2, C) ≅ so(3,1)
sl(2, H) ≅ so(5,1)
sl(2, O) ≅ so(9,1)
Dirac equation is the C case, the other cases have their uses.
The word "quaternion" just rolls of the tongue. I always upvote it.
They sound shiny and mysterious?
I know you know, just practical intuition for 3D graphics in case someone finds it useful:
There's a 1-1 mapping between complex numbers and 2D rotation matrices that only do rotation and scaling. The benefit is that the complex number only has two coefficients, not four like the matrix. Multiplying these complex numbers is the same as multiplying the equivalent matrices. Quaternions are the same idea just in 3 dimensions (so with 3 imaginary units i j k, not just i, one per plane).
Then why not geometric / Clifford algebra?
"Quaternions" definitely sounds shinier and more mysterious than "geometric algebra". Indeed I can’t immediately come up with any math term more shiny and mysterious, except maybe "transcendental", but as a concept transcendentals are much more familiar to most than quaternions.
That quaternions also solve for what we normally have 3D+time for.
And Lewis Carroll (Oxford (Math)), preferred Euclidean geometry over quaternions, for "Alice's Adventures in Wonderland" (1865).
Quaternions:
q = a + bi + cj + dk
-1 = i^2 = j^2 = k^2
> In a quaternion, if you lose the scalar (a) — the "real" or "time" component — you are left with only the three imaginary components (i, j, k) rotating endlessly in a circle.
(An exercise for learning about Lorentzian mechanics, then undefined: Rotate a cube about a point other than its origin. Then, rotate the camera about the origin.)
4D Quaternions (a + bi + cj + dk) are more efficient for computers than 3D+t Euclideans. Quaternions do not have the Gimbal Lock problem that Euclidean vectors have. Quaternions interpolate more smoothly and efficiently, which is valuable for interpolating between keyframes in a physical simulation.
Why are rotations and a scalar a better fit?
Quaternions were published by William Rowan Hamilton (Trinity,) in 1843, in application to classical mechanics and Lagrangian mechanics.
Maxwell's (1861,1862) original ~20 equations are also quaternionic; things are related with complex rotations in EM field theory too. Oliver Heaviside then "simplified" those quaternionic expressions into accessible vectors.
Is there Gimbal Lock in the Heaviside-Hertz vector field reinterpretation of Maxwell's quaternionic EM field theory? Maxwell's has U(1) gauge symmetry.
And then quantum has complex vectors and some unitarity, too
History of quaternions: https://en.wikipedia.org/wiki/History_of_quaternions
Because in 1985 Ken Shoemake dropped the idea like a bomb on the computer graphics industry and it changed the way hackers thought about rotations forever. https://www.ljll.fr/~frey/papers/scientific%20visualisation/...
I mean, there are practical reasons too (which are mostly just isomorphic to the stuff in the paper). But really that's why. It's part of our cultural history in ways that more esoteric math isn't.