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Symmetry 15 - p3m1
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 relfection.

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 wallpaper tilings.

Kali denotes this symmetry by "*333". You can go to now to experiment with symmetry p3m1.

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Page last updated Wed Jul 28 16:36:48 CDT 1999
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