How many rotational matrices are needed in N-space to achieve an arbitrary attitude when allowed to be successively applied?
Each N by N rotation matrix will allow only a rotation in a plane, so only two dimensions may be selected for rotation at a time. What is a smallest set of these matrices which, when used in succession, can achieve any desired orientation in N space?
Assume an N dimensional hypercube with distinct features at every vertex is being rotated on a computer screen under control of $m$ sense switches. The idea is to minimize the number of switches without sacrificing the ability to orient the object, whose projection appears on the screen, in any desired attitude, eventually, by application of this minimal set of rotations.
So far I have $m=N-1$. This is the set comprised of the set of rotations that all include the $x$ dimension. Is there a smaller set? Perhaps not. But the real question is:
If we open it up to allow multiple dimensions being rotated per switch, does that change the answer?