> These discontinuities are not just an artifact of poor implementation; it can be proven that any representation of 3D rotations using only three values must contain discontinuities.
This is a bit pedantic - and the blog post actually does clarify this - but the problem isn't that a 3D representation of representations has "discontinuities" as such, it's that it's not orientable in Euclidean 3D space. It is similar to the Klein bottle - the mathematical description is continuous, but any Klein bottle made of real-world glass has to intersect itself. Or likewise that a Mobius strip can be demonstrated in 3D but can't be built in Flatland without a 3D entity doing the copy-pasting. Reality just has the wrong topology to represent the full group of rotations. Hence the discussion about projective space later in the blog post.
So adding a fourth gimbal is really tantamount to a correctly-oriented embedding of 3D rotations onto a 4-torus (that is, [0,2pi]^4).
Gimbal lock also relates to another issue of continuity, related to the "plate dance."[1] Rotations themselves have a sense of continuity (infinitesimal differences in either the angle or axis of rotation, aka they are Lie groups), but Euler angles fail to respect the equivalent fundamental theorem of calculus: adding a bunch of infinitesimal changes might say you are at the identity rotation according to Euler angles, but in reality you have flipped the meaning of the right-hand rule and the overall state of the system is not at the identity. In a robotics context, the robot's hand might have done a complete rotation, but its arm is twisted without the robot "knowing." I believe robotic arms used to have a serious problem with this, either overrotating and breaking the arm, or swinging dangerously fast in the opposite direction. Using quaternions / a fourth gimbal / etc. there would be a measurable phase or pole indicating the true state of the system, and letting the robot know how to rotate its arm without malfunctioning. So,
like the Apollo mission, the need for a fourth dimension
to keep track of that stuff - and even the quaternion multiplication structure - comes about pretty naturally without ever thinking about abstract math.
[1] https://en.m.wikipedia.org/wiki/Plate_trick