The third symmetry wallpapers the plane with a vertical translation
and horizontal reflection.
We begin with the tile to the left,
translating vertically and reflecting across the red edges.
Imaginge reflecting the entire tiling across any of the vertical
lines. Note that it lands back on itself, and matches up exactly.
Because of this, we say the tiling has reflexive
Suppose we take a tile and its reflection as indicated on the right.
Using this new, larger tile we can wallpaper the plane using the first
symmetry p1 (i.e. just translations). The resulting tiling will match
the pm tiling animated above. In other words, the pm tiling is also
symmetric under translations.
Another thing to notice about tiling pm is that although we used a
square in this illustration, any rectangle will work.
Kali denotes this symmetry by "**". You can go to now to experiment with symmetry pm.