The earlier post about commutivity and turning the book was defined by operations in eucledian space. I suggest that the reason that the operations were not commutative was the choice of a rectangular coordinate system not appropriate for the discussion. Instead we should consider a spherical geometry when discussing operations on angles in three dimensions.
Consider the earth. Let us take the north pole as our origin. If we take an equatorial axis such that any positive angle describes a line down the prime meridian, and we rotate through 130 degrees we get to 40 degrees south, in the region of South Africa. If we then rotate along the polar axis by 174 degrees we get to New Zealand. Now this second rotation has not changed the arc length as the resulting arc is from the North Pole to New Zealand. This is because we are working off a sphere.
What now happens if we reverse the order of operations? Standing on the North Pole, we are rotated along the polar axis, next we rotate along a different meridian and end up at New Zealand. We have our commutivity back by changing our coordinate system. If you try this with the book it works, the book faces the same way after both rotations.
