actually, this turned out to have more questions than I supposed. so I'm gonna number these for clarity:
1: although it is generally assumed that rotation in 3d happens around an axis, leaving it invariant, there also exists what is called 'spherical rotation' in 3d : rotating around an axis A, while simultaneously rotating this rotation axis A around another axis B, gives spinning motion - so called spherical rotation. then ( speed of rotation on A / speed of rotation on B ) & (angle between A and B) determines the kind of loop (if any) one gets. does this leave a point of intersection of A and B invariant? or not even that (because it would get somehow 'twisted' i imagine) ?
2: then, in 4d, rotating for exemple around plane(xy), while simultaneously rotating this plane(xy) around another plane(wz), what do we get - a 'glomar rotation' ?
3: then, what (if any) influence does it have in 4d, whether 1 of 2 coordinates of the first rotation plane is the same as one of second rotation plane's 2 coordinates ?
I mean: is there a difference between 'glomar rotations' for these 3 couples of rotation planes: [(xy)(zw), (xz)(yw),(xw)(yz)] and these 12: [(xy)(xw), (xy)(xz), (xy)(wy), (xy)(zy), (xz)(xw), (xz)(zw), (xz)(zy), (zw)(wx), (zw)(wy), (zw)(zy), (yw)(wx), (yw)(yz) ] ?
{btw, can anyone give a lesson in notation ? i'm not mathematician, but surely there must be a concise way to note the above questions... }
4: finally, how to generalize this process ? firstly its iterative character:
r0 - rotate something (= ordinary rotation around axis)
r1(r0) - rotate this rotation (= 'spherical rotation', 'spin', 'double rotation')
r2(r1(r0)) - rotate the spherical rotation (= 'triple rotation')
rn(...r3(r2(r1(r0)))...) - 'n-tuple rotation' ???
r(infinity) - 'infinity-tuple rotation ???
5: then to understand it across increasing dimensionality...
6: or, because of some (to me yet not understood) reasons, this sequence of 'rotational iterations' breaks down and stops ? (like for exemple the 4 divisor algebras end up at octonions), or does something like the Gimbal lock in 3d, occuring recursively, forces the whole thing into stasis ?
.done