This symmetry contains horizontal translations and our first encounter
with glide reflections.
The animation uses the pg symmetry group to wallpaper the plane
with the tile shown to the left.
Glide refection shift the tile horizontally and reflect over the
red lines.
Notice if we glide reflect the entire tiling along these red lines
(i.e. reflect across and then shift half a tile width) the entire
tiling lands on top of itself and matches up. When this happens, we
say a tiling is symmetric with respect to glide reflections.
Using glide reflections, we
can construct the larger tile shown to the right out of four copies of
the original tile. Can you locate some instances of this larger tile
in the pg tiling on the left? If you look at the rectangle outlined in
black, notice you can use just translations
(i.e. the first symmetry group p1) to
wallpaper the plane with this rectangle, and the resulting tiling is
the same as the pg tiling. So the pg tiling is also
symmetric with respect to translations.
Notice that, although our illustration uses a square, any rectangle
will work.
Kali denotes this symmetry by "oo". You can go to now to experiment with symmetry pg.
