The seventh symmetry wallpapers the plane with half turns (rotations
of 180 degrees) and reflections.
We begin with the tile shown to the left. Rotate about the red points, and
reflect across the vertical red lines. Although we are illustrating
the symmetry with square, any rectangle could be used to wallpaper the plane.
If we reflect the entire tiling about any of the vertical lines
then the tiling lands back on itself in such a way that it matches up
exactly. Also, if we rotate (by 180 degrees) the tiling about any of
the red points, it matches up with itself. Because of this, we say
this tiling has reflexive and two-fold rotational symmetries.
Suppose we reflect and
rotate our original tile to obtain the larger rectangle shown to the
right. As a rectangle, it will wallpaper the plane under the p1
symmetry group, i.e. the translations. This new tiling
will agree with that above. Therefore, pmg tilings are also
symmetric with respect to translations.
Kali denotes this symmetry by "22*". You can go to now to experiment with symmetry pmg.