Number of components needed for 3D rotation Using Euler angles, a 3D rotation can be expressed using 3 real numbers. Using quaternions, 4 are needed and using rotation matrices 9.
Is it possible to express a 3D rotation using less than 3 real numbers, or why are 3 needed?
 A: No, it's not possible, at least not possible in a bi-continuous way. 
Because by a generalization of a Peano curve, there's a map from the unit interval onto the unit cube in 3-space, the euler-angle parameterization can be reduced to a single-number parameterization...but that's completely useless in practice. 
Now...why, if you want a "nice" parameterization, do you need three coordinates? 
Well, start at the "no motion" rotation, and consider the three independent rotations about the x, y, and z axes. What I mean by "independent" is that no small rotation about x -- say, less than 10 degrees -- can be expressed as a combination of small rotations about y and z. 
On the other hand, every small rotation  (e.g., less than 5 degrees) can be expressed as a combinations of small-ish (less than 10 degrees) rotations about x, y, and z. 
In short: there's a 3-parameter nonsingular parameterization of a small region of rotation space. Let's call this 
$$
H: I \times I \times I \to SO(3)
$$
If you want to parameterize rotations near some rotation $R$, you can use
$$
H_R (a, b, c) = R \circ H(a, b, c).
$$
So there's a nonsingular 3-parameter parameterization of a neighborhood of any rotation $R$. 
That makes the set of rotations -- let me call it $SO(3)$ -- into a topological 3-manifold. And the Brouwer theorem on the invariance of domain then shows that any other "nice" parameterization that's 1-1 (as $H$ is) must also have three parameters. 
