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.

[1] https://arxiv.org/abs/math/0105155

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

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.

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.

Also lots of HN readers are actual physicists or mathematicians. It's not all techies.

Also, lots of engineers have at some point learned some computer graphics and so been exposed to quaternions in that setting. Since they're mysterious and hard to wrap your head around most people don't really 'get it', leaving a sort of standing curiosity that articles like this tap into.

That's one of those things I've vaguely learned but really want to spend the time to learn academically so I can actually do the math instead of just hearing about it.

A 2d quaternion just has no j or k term and works for 2d math.