A book on algebra by Sullivian contains a sentence like this:
Symmetry with respect to the origin may be viewed as a reflection about the y-axis, followed by a reflection about the x-axis.
Here,what does the phrase "reflection about" means? is it equal to the phrase "reflection on"?
Here, about takes the following meaning listed at Merriam-Webster:
around the outside
In this context, the sentence refers to a reflection of what is on one side of an axis, to the opposite side, or "outside" of the axis.
This could also be described as a reflection around an axis:
in or to an opposite direction or position
Also, across an axis:
from one side to the opposite side of
To describe a reflection as being on an axis would be less clear than either of the above terms, as it could also indicate a reflection of the content found on one side of an axis back onto that same side.
However, a reflection in an axis would work, in the same way that a reflection in a mirror refers to a (conceptual) "other side" across the plane of reflection.
A useful example of the overall meaning from Wikipedia reads:
... the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b.
