This symmetry group is made up of reflections and glide reflections with
We begin wallpapering with the tile on the left. Reflect along the
red edge of the triangle and glide reflect along the black arrows
indicated in the middle of the triangle.
Notice if we glide reflect the entire tiling as indicated by any of
the black arrows the resulting tiling matches the original perfectly.
We witness a similary phenomenon when reflecting the tiling across the
horizontal lines of reflection (shown in red on the original tile).
Because of this, we say the tiling is symmetric with respect to
reflections and glide reflections.
Suppose we contruct the tile on the right by reflecting the original
tile across its red edge. We can then wallpaper the plane with this
parallelogram using only translations, i.e. the
p1 symmetry group. The resulting tiling will be the same as the one
animated above using the cm symmetry group with the original tile. As
a consequence, we see the cm tiling is also symmetric with respect
Kali denotes this symmetry by "*o". You can go to now to experiment with symmetry cm.