This is another symmetry group based on the p3
symmetry group (p3 is generated by 120 degree rotations). Like pm31, this symmetry group also adds reflections
(with axes at 60 degrees to each other). Unlike the pm31 symmetry,
however, all the rotations are at the points where the
reflection axes intersect.
For this kind of tiling, we begin with the obtuse triangle to the left
as our fundamental domain. We then rotate about the blue/green vertex
and reflect across the red edge.
To see the reflexive and 3-fold
symmetry of this tiling,
imaging rotating the entire tiling about one of these blue/green
vertices. The the rotated tiling matches the original perfectly,
as if we had never rotated the tiling. The same thing happens
when we reflect the tiling across any of the red lines of
Using the p3m1 symmetry group, we can construct the hexagon to the
right from our original tile from above. You can recreate the whole
tiling by translating around the hexagon, which means p3m1
tilings are periodic, like all
Kali denotes this symmetry by "*333". You can go to now to experiment with symmetry p3m1.