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Symmetry 17 - p6m
The seventeenth and final symmetry is the most complicated of all the wallpaper symmetry groups. It contains the 2-, 3- and 6-fold rotational symmetries that p6 has, and in addition, it has reflections.

The p6m tiling animation on the right starts with the triangle to the left. This is a right triangle with 30 and 60 degree angles (in addition to its 90 degree "right" angle). The p6m symmetry group allows us to reflect across all sides of the triangle. In addition, we can rotate by 60 degrees about the vertex with the 30 degree angle. Similarly, we can rotate by 120 degrees about the 60 degree vertex and rotate by 180 degrees about the right (90 degree) vertex. However, as the animation illustrates, we don't actually have to use any of the rotations -- we can wallpaper the plane with just the refections.

Since we can reflect the entire tiling through any of the red lines and have it match back up with itself, we say this tiling is symmetric under reflections. Similarly this tiling contains 6-, 3- and 2-fold rotational symmetry about those points where or 6, 3 or 2 lines of reflection intersect. If you know about isometries, you will see this happens because doing two intersecting reflections is the same as a rotation.

As with the p6 tiling, we can choose a hexagonal fundamental domain like the one shown to the right, and produce the original p6 tiling by translating it around. There is always such a fundamental domain for wallpaper tilings, since they are periodic.

Kali denotes this symmetry by "*632". You can go to now to experiment with symmetry p6m.



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