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Symmetry 5 - cm
This symmetry group is made up of reflections and glide reflections with parallel axes.

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 to translations.

Kali denotes this symmetry by "*o". You can go to now to experiment with symmetry cm.



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