What is a bilateral binary rotation?

[TimH]

I may be posting to the wrong forum, if so, please advise. I'm reading a book on rotational symmetry and its relationship to quantum mechanics. The author is talking about rotations of a unit sphere, in particular binary rotations, i.e. rotations of 180 degrees. He says "If a rotation [he then uses notation to specify it need not be binary] has a binary axis perpendicular to it, then the two semiaxes are interchanged by the binary rotation."

I don't understand this. Okay, we have a non-binary rotation, say a rotation in the x-y plane. The z-axis is then an axis perpendicular to this rotation. We can make a binary (= 180 deg) rotation around z. What does it mean to say the "semiaxes are interchanged?" I've googled "semiaxes" and it refers usuall to the axes of an ellipse. What does it mean here?

He then says that if the first rotation is itself binary, then the rotation is called a "bilateral binary rotation."

Any help appreciated.

 

[HallsofIvy]

I don't think the problem here is mathematical. I'm moving this to "quantum physics".

 

[TimH]

This is not really a question about QM, its about the mathematics and terminology used to describe rotations in the study of the rotation group. The book (Rotations, Quaternions, and Double Groups, by Altmann) eventually gets to QM and symmetry in molecules. The author in the quote (above) is saying something about how if you have an angular rotation(say of the unit sphere in R3), and you rotate the whole sphere by pi in a plane perpendicular to the rotation, you switch the "semiaxes" of the rotation. I'm just trying to visualize this Anybody familiar with symmetry/rotation issues in molecules might be familiar with this sort of visualization and terminology. Thanks.

 

[quaternion2]

Altmannn is a a bit confusing here, but not to worry, this is all about definitions. Consider any rotation about, say, the z-axis. Consider another rotation axis, labeled the x-axis, perpendicular to the z-axis (and which bisects the z-axis--each part of which is called a "semi-axis"). Now, a binary rotation about the x-axis interchanges these two semi-axes. Draw it out if you cannot visualize it. Here is the confusing definition: with such a perpendicular binary rotation with respect to the z-axis, rotations about the z-axis are called "bilateral rotations." If these z-axis rotations are themselves binary, then they are called "bilateral binary rotations". And now, since the binary rotation about the z-axis interchange the x semi-axes, rotations about the x-axis are now, using the same definition, called bilateral binary rotation. Thus it can be seen that "...bilateral binary rotations must always appear as pairs of mutually perpendicular binary axes." Hope that helps...