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Symmetry 8 - pgg
The eighth wallpaper symmetry group is made up of glide reflections and 180 degree rotations (half-turns). There are two glide reflections and they are perpendicular to each other. Although the example shown here uses a square tile, the pgg symmetry will wallpaper the plane with any rectangle.

Tilings with pgg symmetry are constructed as follows. Begin with the tile on the left. Then build up a tiling by rotating about the red points at the top and bottom of the square, and applying glide reflections, as shown in the animation. The axis of the first glide reflection is along the horizontal red line through the middle. For the axis of the second glide reflection, you can use either vertical edge.

Notice if we rotate the entire tiling (by 180 degrees) about any of the red points then it lands back on itself, matching the original tiling perfectly. Hence, this tiling has two-fold rotational symmetry. Similarly if we glide refect the tiling along any of the horizontal or vertical lines it lands back on it self and matches up, so we say the tiling is symmetric with respect to the glide reflections.

We can rotate and glide reflect our original tile from above to build the large square on the right. This new square, built from four copies of the original tile, wallpapers the plane using just translations, which is the p1 symmetry group. Since the new tiling constructed from the larger tile matches the original pgg tiling, we see that pgg tilings are symmetric under translations. This is another way of saying it is a periodic tiling.

Kali denotes this symmetry by "22o". You can go to now to experiment with symmetry pgg.



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