Is there a general mathematical relationship between this cyclic behavior and infinite series?

I think it is not uncommon that one series can be separated usefully into two or more based on some alternating / cyclic behavior.

I found the mathematical relationship between all these seemingly different spaces to be quite beautiful.

A good way to gain some physical intuition of why the exponential map encodes motions in space is to think of the maths in relatively simple situation. At an extremely high level:

1. Recall some basic details: a) that the solution to ODE $\dot x(t) = ax(t)$ is $x(t) = e^{at}x_0$, with initial condition $x(0) = x_0$ b) that the velocity of a point $q$ (i.e. the time derivative of the position of point $q$) rotating (and not translating) in space is given by cross product of the angular velocity vector and the vector $r$ providing the position (from the axis of rotation)
2. We use the hat product to “convert” the linear velocity vector and cross product to a matrix: we can convert this cross product into a matrix-vector multiplication instead, since the cross product is a linear transformation. The angular velocity vector can be converted into a matrix using the hat map operator. This new matrix is such that when any vector (in this case the $r$ vector) is applied to it, it is equivalent to the taking the cross product.
3. Notice, therefore, that the position of the point of a point $q$ is therefore given by $e^{\hat wt}x_0$, with initial condition $q(0) = q_0$ (because it is the solution to the ODE). (Where $\hat w$ is the hat map of the angular velocity vector)

This is a fairly simple scenario to follow in your head but provides good physical intuition of why the exponential map encodes motions in space: it is the result of a solution to an ODE that relates a rotating body’s velocity to it’s position. This doesn’t go into any of the finer details or interesting mathematical proofs. The very interesting maths comes in later: finding a closed form for the exponential map—which results, eventually, in the Rodrigues’ Formula.

@chetan-khanna Excellent notes! Yes, the ODE interpretation is the most intuitive in my opinion. Although, with a beautiful formula like this, I am sure there are other interpretations that I am not aware of.

(Aside: to get the LaTeX formatting to work correctly, use single $ instead of double $$.)