Symmetry 10 - p4
The tenth wallpaper symmetry group uses rotations of 90 and 180 degrees to tile the plane.

To construct a p4 tiling beginning with a tile like that shown on the left, do 90 degree rotations about the red corner (upper right), and 180 degree rotations about the yellow corner (lower right).

A p4 tiling is symmetric under two- and four-fold rotations. To see this, imagine rotating the entire tiling 90 or 180 degrees about any red or yellow point, respectively. The rotated tiling will match up with the original perfectly, which is just what it means to have rotational symmetry.

By rotating the original tile three times about the red point, we can construct the square shown on the right. Using the just href="p1.html">translation (i.e. the first symmetry group p1) we can tile the plane with this new square. It is easy to see that this new tiling will match the p4 tiling shown in the animation above. Therefore, p4 tilings are symmetric with respect to translations as well. This is just another way of saying that this is a periodic tiling.

Kali denotes this symmetry by "442". You can go to now to experiment with symmetry p4.

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