Wendell gets at the bit I'd have skipped straight to — four numbers. That's it. One real, three that aren't, and you can write the whole attitude of a spacecraft on the back of a receipt and still have room for the shopping. Three numbers, the roll-pitch-yaw way, sounds tidier until two of your axes line up and the thing loses a whole dimension of knowing where it's pointing. Gimbal lock. Apollo's platform had a no-go zone the crew had to steer around, and there's a reason the flight controllers made grim jokes about it.
Quaternions don't have that hole in them. You can spin a star tracker any which way, hand it the four numbers, and it just keeps counting. Hamilton was out for a walk with his wife, 1843, thinking about algebra, and the arithmetic he scratched onto that parapet is running right now in something a few hundred kilometres up, quietly keeping an antenna aimed at Earth.
I was cleaning out a drawer of junction-box spares when I thought about Hamilton and his penknife, scratching i\u00b2 = j\u00b2 = k\u00b2 = ijk = -1 into the parapet of Broom Bridge in 1843 because he hadn't any paper and didn't trust…
A man scratches something on a bridge with a penknife and a century and a half later it's in the flight software, which is more than most of my solder joints ever managed. Lyra, I checked — the carving's long gone, the plaque's the replacement.