I'm reading a magazine article on 3d orientation and want to know what the mathematical issue is and where to read about it, preferably a theorem title that I can google or a topic and textbook:
It’s possible to prove that no three-scalar parameterization of 3D orientation exists that doesn’t suck, for some suitably mathematically rigorous definition of “suck.” I haven’t done this proof (I think it uses some pretty high-end group theory, which I haven’t learned yet), so I can’t tell you exactly how it works, but I believe the gist of the proof is that no minimal parameterization exists that doesn’t contain singularities. These singularities can take different forms — depending on how you allocate the three degrees of freedom — but according to the math, it’s impossible to get rid of them.
3blue1brown glazes over it by saying "As it happens, it's not possible to define a notion of multiplication for a 3d numer system like this that would satisfy the usual algebraic properties that make multiplicatiohn a useful construct" but goes no deeper. Saying he may make a future video. What is a good resource to understand the issue?
It is because the 3D orientations are, in their essence (i.e. topologically), equivalent to the parameterization of a circle and a sphere. You can see this in some of the representations such as the axis-angle or the quaternions. But no matter the representation, to specify an orientation, you need to choose a point on a circle and a point on a sphere. Sometimes it is not always clear that that is what you are doing but under the covers a rotation always chooses a point on a circle and a point on a sphere and with any representation you can always get back to these points. The problem is then the parameterization of the sphere. As mentioned in the comments, it is not possible to parameterize the sphere with two parameters and not have at least two singular points (i.e. hairy ball theorem). Fortunately you can parameterize the sphere with three points and one constraint (i.e. a vector of unit length).