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Symmetry 9 - cmm
The ninth, or cmm symmetry contains perpendicular reflections and half-turns (180 degree rotations).

To build a cmm tiling, begin with the tile on the left. Rotate the tile about the red point and reflect it across the red lines.

If we rotate the entire tiling 180 degrees about any of the red points the rotated tiling will match the original tiling perfectly. Because of this, we say the tiling contains two-fold rotational symmetry. If you look closely you will see that the tiling also lands back on top of itself if you rotate by 180 degrees about points where the vertical and horizontal lines of reflection intersect. Those of you who know a bit about isometries you will realize that this is because each rotation is a composition of reflections.

The cmm tiling shown above is also symmetic with respect to reflection. That is if we reflect the entire tiling about one of the horizontal/vertical lines of reflection then we get the same tiling again.

Suppose we reflect the original tile from above to obtain the larger rhombus shown to the right. Since a rhombus is a parallelogram we can use the p1 symmetry, that is just the translations, to wallpaper the plane with this new tile. This new tiling will match the original cmm tiling, and so cmm tilings are also symmetric with respect to translations.

Kali denotes this symmetry by "2*22". You can go to now to experiment with symmetry cmm.

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Page last updated Wed Jul 28 16:36:47 CDT 1999
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