Symmetry 16 - p6 (Science U)

Symmetry 16 - p6

The sixteenth symmetry group is notable for containing 60 degrees rotations. Since a 60 degree rotation is a sixth of a full turn, p6 tilings have 6-fold symmetry. This means these tilings must also have 2-fold and 3-fold symmetry. To convince yourself of this, remember that having 6-fold symmetry means that there are places in the tiling where we can rotate it by 60 degrees and have the tiling match back up with itself. But if we just rotate two or three clicks at a time, we will come back to the original position after either three or two times, i.e 3-fold and 2-fold symmtery respectively. All of this makes for a very symmetric tiling!

In the animation, we begin with the triangle to the left, and then rotate by 60 degrees about the blue/green vertex, by 120 degress about the solid blue and green vertices, and by 180 degrees about the red point. If we rotate the entire tiling about any of these points (by the appropriate amount) then the rotated tiling matches up with the original. In other words, these are the specific points that exhibit the 2-, 3- and 6-fold symmetry of the tiling.

Since all wallpaper tilings are periodic, we know that it must be possible to build up a p6 tiling from a fundamental domain, just by using translations. One possible fundamental domain is the hexagon to the right, constructed out of six coppies of the original tile.

Kali denotes this symmetry by "632". You can go to now to experiment with symmetry p6.