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Symmetry 7 - pmg
The seventh symmetry wallpapers the plane with half turns (rotations of 180 degrees) and reflections.

We begin with the tile shown to the left. Rotate about the red points, and reflect across the vertical red lines. Although we are illustrating the symmetry with square, any rectangle could be used to wallpaper the plane.

If we reflect the entire tiling about any of the vertical lines then the tiling lands back on itself in such a way that it matches up exactly. Also, if we rotate (by 180 degrees) the tiling about any of the red points, it matches up with itself. Because of this, we say this tiling has reflexive and two-fold rotational symmetries.

Suppose we reflect and rotate our original tile to obtain the larger rectangle shown to the right. As a rectangle, it will wallpaper the plane under the p1 symmetry group, i.e. the translations. This new tiling will agree with that above. Therefore, pmg tilings are also symmetric with respect to translations.

Kali denotes this symmetry by "22*". You can go to now to experiment with symmetry pmg.



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