alt Hacker NewsI 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?
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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.
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.
The word "quaternion" just rolls of the tongue. I always upvote it.
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.
Honestly, with all my love for the HN community, I think we have a couple of topics that just get upvoted without reading because they signal that you're in the ingroup. Few years back, another reliably upvoted thing was anything with "Bayesian" in the name. In the past couple of years, "busy beavers" would also get upvotes even though they have no practical use, their mathematical significance is dubious, and few people understand them in the first place.