The sixth symmetry consists of two perpendicular
reflections. We begin with the rectangular tile shown to the left and
proceed to reflect across the edges. Although in this example we use
a square, any rectangle would work.
If we reflect the entire tiling across any of the horizontal or
veritcal lines then the tiling lands back on itself in such a way that
it matches up exactly. Thus, this tiling has reflexive symmetry.
Now take the original tile from above and reflect three times to obtain the
larger square shown to the right. We could use the
translations to wallpaper the plane with this
new tile, and we would the same tiling as
that generated above using the pmm symmetry group with the original
tile. Therefore, pmm tilings are also symmetric with repect to
translation. This is the same as saying pmm tilings are periodic.
Kali denotes this symmetry by "*2222". You can go to now to experiment with symmetry pmm.
