# Confusion about bilateral binary rotations

I'm reading a book called 'Rotations, Quaternions, and Double Groups' and I'm having a great deal of difficulty trying to understand the definition of bilateral rotations, bilateral binary rotation, and semiaxes.

Here is the snippet I'm confused about,

"Binary rotations and their corresponding axes, or binary axes, will be found to be most important in the study of the rotation group. One particularly significant property which they have is this: if a rotation $C_m$ has a binary axis perpendicular to it, then the two semiaxes of $C_m$ are interchanged by the binary rotation"

1) What are the semiaxes of $C_m$? My guess is if we take the +Z to be the binary rotation axis and $C_m$'s rotation axis to be +X then I'm guessing when he says semiaxes he must mean +X and +Y? I can't find any good google results for semiaxes. If so when Altmann says interchanged he means to say that +Y and +X will now be pointing in the opposite directions, correct?

He continues on with, "Rotations $C_m$ thus related to a binary rotation are called for this reason bilateral rotations."

2) What does mean when he says 'thus related to a binary rotation'. I understand what he means after a few more readings

To finish the paragraph he goes on to say, "If the rotation $C_m$ here is itself binary, $C_2$, then it is called a bilateral binary rotation. It is clear in this case that the second $C_2$ is also bilateral binary; that is, that bilateral binary rotations must always appear as pairs of mutually perpendicular binary axes."

Thank you, I've been having a hard time finding other sources of information on this so I'm quite confused on his definitions.

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From what you quote, "if a rotation $C_m$ has a binary axis perpendicular to it, then the two semiaxes of $C_m$ are interchanged by the binary rotation", it seems that by a semiaxis they mean the two rays making up an axis -- so in your example, if the binary rotation axis is the $Z$ axis and $C_m$ is a rotation about the $X$ axis, then the semiaxes would be the $+X$ ray and the $-X$ ray.