The final wallpaper groups allow for rotations of 120° (a third of a circle) or 60° (a sixth of a circle). I’m presenting these groups together since I find it helpful in each case to think of the diamond-shaped tile that you can identify in the pattern. That will make more sense after the first example!
Group p3
Patterns in this group have centers of rotation of 120° but no reflections. This isn’t a pattern group I have stumbled upon, so I reused my moon and sun motifs to try it out. The center of each sun is a center of rotation. The center point between the trio of half moon shapes and the center point between the trio of dots are also both rotation points.
The image on the right shows three of the tiles outlined. Each vertex is one of the centers of rotation.
Group p3m1
This next group has reflections. The diamond-shaped tile is symmetrical along its shorter diagonal (along the dashed blue line). If you extend those dashed blue lines, they are reflection lines for the entire pattern. All of the centers of rotation are at the intersections of the reflection lines.
Group p31m
In this group, we’ll also start with a diamond-shaped tile, but this time it will be symmetrical along the longer center line. Here I’m using motifs from my aster checkerboard pattern (see a scarf I made from this pattern on the right). I love how different the effect is from just changing the arrangement of elements!
I’ll show another couple of example from this group of collages I made. The key to distinguishing p31m from p3m1 is that p31m has centers of rotation that aren’t along the reflection lines.
It took me a bit to figure out what group these patterns belonged to. First I drew the reflection lines, and then identified the centers of rotation. One of the centers of rotation (the center point of the three tiles outlined) is not on a reflection line (the dashed blue lines), so that told me this pattern belonged to group p31m. Only after drawing the reflection lines and rotation points could I figure out what the diamond-shaped tile was.
Group p6
The p6 pattern group has centers of rotation of 60°, but no reflections. I find it helpful to keep thinking of the diamond-shaped tile. The previous three groups had diamond-shaped tiles that had no reflection lines (p3), a reflection along the short diagonal (p3m1), or a reflection along the long diagonal (p31m). The diamond-shaped tile in p6 doesn’t have any reflection lines, but it does have rotational symmetry of 180°.
An example for this group is below. Each diamond has diagonal lines, so the diamond doesn’t have reflections down its diagonals. You can rotate each diamond 180° and get the same motif.
Group p6m
Last but not least, group p6m! If we think about the diamond-shaped tile again, this group’s tile has reflection lines along both the short and long diagonals. The final pattern has centers of rotation of 60° (a sixth of a circle) and it has reflections. The example I have for this group is actually one of my favorite collages! It’s hanging up in the entryway of my apartment. This was one of the first pieces where I painted a floral pattern and then cut it up to create a geometric collage. Only after working on this blog series did I realize it’s an example of the p6m wallpaper group.
Conclusion
And that’s it! In these five blog posts we’ve covered all 17 types of repeat pattern symmetry. What I love about studying these groups is it helps me expand my pattern ideas. I often create patterns that have fairly simple symmetry, like p1 or cm. When I force myself to try a different symmetry group, I’m often surprised and pleased by the result! For example, when I first designed my starry sky pattern, I just scattered the elements around a rectangle so it was a p1 pattern. Then I intentionally tried it as a pgg pattern, and I think the result is much more compelling.
The repeat aspect of patterns is what draws our attention. A complex repeat isn’t always better, but I think it’s worth experimenting with different symmetries to develop engaging and interesting patterns.
To learn more about the wallpaper groups, the wikipedia article is quite good and provides visual examples of each group! https://en.wikipedia.org/wiki/Wallpaper_group
Another good website is from a math professor at Clark University: https://www2.clarku.edu/faculty/djoyce/wallpaper/