To study mirror symmetry at the x-axis together with rotational symmetry we can do similarly as in the earlier post “improved symmetric sum“. Here I prefer to present only the conclusions, which you could get by intuition too.
It is important to know how the packages of wave functions with rotational symmetry change when the drawing plane is mirrored. What is
for r-fold rotational symmetry and where M stands for a mirror symmetry?
Note that we want a mirror symmetry at the x-axis with M(x,y)=(x,-y). Then the rotational symmetry gives us more mirror symmetries at axis with angles of integer multiples of π/r. Actually, the unit vectors e are mirror axis and the lines exactly between two adjacent unit vectors are mirror axis too. M can be any of these mirror symmetries. Then
where the mirror image of the wave vector simply reverses its components. In general:
where we can do additional, essentially irrelevant rotations. If the wave vector has only one non-zero component then the package of wave functions is mirror symmetric, M(k,0,0,…)≅(k,0,0,…). If there are two non-zero components, then they get exchanged, M(k,g,0,0,…)=(g,k,0,0,…) and we can get a wave package that is not mirror symmetric with respect to the x-axis.
We can always make that a rotationally symmetric package of waves becomes mirror symmetric by adding its mirror image
but most of the time I get a mirror symmetric image by accident.
Actually, it is quite difficult to get quasi-periodic images that are not mirror symmetric. Here is an example of 5-fold rotational symmetry. The center of perfect symmetry lies near its bottom left corner.
Look out for local approximate mirror symmetries !