Suppose I want to create an icosahedron by building a set of twenty triangular pyramids (aka tetrahedrons, but see below) of an appropriate size and then rotating each into position. I need to do three rotational transforms: the first, $\tau$, tau, around the central (X-)axis of the pyramid to get the correct "twist"; the second, $\theta$, around the Y-axis to get the correct elevation; then the third, $\phi$, around the Z-axis to get the right azimuth. The figure below shows an example of what I'm trying to do.
As the image shows, I can find $\theta$ and $\phi$, but I can't find a way to calculate $\tau$. (And, in fact, it's wrong in the image.) I can keep guessing values and get close, but I'd rather have a formula so I can get it exactly. PLUS, I'd like to be able to do the same thing with the other Platonic solids, making a formula even more necessary.
So, in a nutshell, the problem is: how to calculate $\tau$ given, say, the vertices of a desired face and the vertices of a constructed pyramid. (FWIW, I calculated $\theta$ and $\phi$ by calculating the centroid of the face and then converting from Cartesian to polar coordinates.)