The second symmetry wallpapers the plane with a horizontal translation
and a 180 degree rotation. This animation wallpapers the plane with
the tile to the left. We rotate about the red points indicated on the tile.
Observe that if we rotate the entire tiling 180 degrees about any
of the red points, the tiling lands back on itself. We say the
tilings is symmetric with respect to 180 degree turns around any
of the red points.
Suppose we take the tile at the top left and stack one on top of the other as
indicated to the right. The we can use the first symmetry - p1 - to
wallpaper the plane with this larger tile. Moreover, we will obtain
the same tiling of the plane as the p2 animation above is
constructing. Hence, the p2 tiling is also symmetric under two
translations. (Can you find the two translations?)
Kali denotes this symmetry by "2222". You can go to now to experiment with symmetry p2.