To jump the gun slightly on the group-theoretic explanation,
any sequence of rotations of any number of faces can be
thought of as an atomic "transformation" for the purposes of
group theory. One of the precepts of this theory is that any
such transformation, repeated often enough, will return the
cube to the original state. For instance, given the transform
"rotate top ccw 90", 4 iterations suffice to return the cube
to the original state.
Mike Speciner, a fellow Camexian, claims that no transformation
can be created that requires more than 216 (=6^3) iterations to
return to the virgin state. (He doesn't yet have a cube, but
has been stealing his daughter's blocks and modeling the cube
with them.)
Where does this number come from, and is it true?
I have been playing with various transforms, and have found
at least one reasonably trivial one that requires the 216
iterations: rotate a face 90, then turn the cube 90 and repeat.
The transform here is 4 twists in a band around the axis of
cube rotation. The patterns generated in the process are interesting,
too, though none of them are as unique as the cruciform or center-face patterns.